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All Textbook Solutions for Precalculus
A system of linear equations in two variables may have no solution. In such a case the equations represent lines.A system of equations that has no solution is called an system.A system of linear equations in two variables may have infinitely many solutions. In such a case, the equations are said to be .For Exercises 7-10, determine if the ordered pair is a solution to the system of equations. (See Example 1) 3x5y=7x4y=7 a. 1,2 b. 23,1For Exercises 7-10, determine if the ordered pair is a solution to the system of equations. (See Example 1) 11x+6y=47x+3y=23 a. 1,76 b. 2,39PEFor Exercises 7-10, determine if the ordered pair is a solution to the system of equations. (See Example 1) y=15x+22x+10y=10 a. 5,1 b. 10,4For Exercises 11-14, a system of equations is given in which each equation is written in slope-intercept form. Determine the number of solutions. If the system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. y=25x7y=14x+712PE13PEFor Exercises 11-14, a system of equations is given in which each equation is written in slope-intercept form. Determine the number of solutions. If the system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. y=12x+3y=2x+12For Exercises 15-20, solve the system of equations by using the substitution method. (See Example 2) x+3y=53x2y=18For Exercises 15-20, solve the system of equations by using the substitution method. (See Example 2) 2x+y=25x+3y=917PEFor Exercises 15-20, solve the system of equations by using the substitution method. (See Example 2) 3x=2y116+5x=1For Exercises 15-20, solve the system of equations by using the substitution method. (See Example 2) 2x+y=2y4x1=25yFor Exercises 15-20, solve the system of equations by using the substitution method. (See Example 2) 5x+y=9+2y6y2=107xFor Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 3x7y=16x+5y=17For Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 5x2y=23x+4y=30For Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 11x=54y2x2y=22+yFor Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 3xy=y142x+2=7yFor Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 0.6x+0.1y=0.42x0.7y=0.3For Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 0.25x0.04y=0.240.15x0.12y=0.12For Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 2x+11y=43x6y=5For Exercises 21-28, solve the system of equations by using the addition method. (See Examples 3-4) 3x4y=92x+9y=2For Exercises 29-34, solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples 5-6) 3x4y=69x=12y+4For Exercises 29-34, solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples 5-6) 4x8y=22x=84yFor Exercises 29-34, solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples 5-6) 3x+y=6x+13y=2For Exercises 29-34, solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples 5-6) 2xy=8x12y=4For Exercises 29-34, solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples 5-6) 2x+4=45y2+4x+y=7y+2For Exercises 29-34, solve the system by using any method. If a system does not have one unique solution, state whether the system is inconsistent or whether the equations are dependent. (See Examples 5-6) 3x3y=2y2x+5=57yFor Exercises 35-36, a. Write the general solution. b. Find three individual solutions. Answers will vary. 5xy=610x=2y+6For Exercises 35-36, a. Write the general solution. b. Find three individual solutions. Answers will vary. 2y=64x8x=124yFor Exercises 37-50, solve the system using any method. 3x10y=19005y+800=xFor Exercises 37-50, solve the system using any method. 2x7y=24004+1800=yFor Exercises 37-50, solve the system using any method. 52x+y=yx8x32y=52For Exercises 37-50, solve the system using any method. 32xy=2xx+54y=32For Exercises 37-50, solve the system using any method. y=23x1y=16x+2For Exercises 37-50, solve the system using any method. y=14x+7y=32x+17For Exercises 37-50, solve the system using any method. 4x2=6y+314x38y=12For Exercises 37-50, solve the system using any method. 114x17y=122x2y+3=20For Exercises 37-50, solve the system using any method. 2x=y2+10.04x0.01y=0.02For Exercises 37-50, solve the system using any method. 0.05x+0.01y=0.03x+y5=35For Exercises 37-50, solve the system using any method. y=2.4x1.54v=3.5x+7.9For Exercises 37-50, solve the system using any method. y=0.18x+0.129y=0.15x+0.1275For Exercises 37-50, solve the system using any method. x28+y+12=6x22y+14=12For Exercises 37-50, solve the system using any method. x+12y210=1x+16+y22=21One antifreeze solution is 36 alcohol and another is 20 alcohol. How much of each mixture should be added to make 40L of a solution that is 30 alcohol? (See Example 7)A pharmacist wants to mix a 30 saline solution with a 10 saline solution to get 200mL of a 12 saline solution. How much of each solution should she use?A radiator has 16L of a 36 antifreeze solution. How much must be drained and replaced by pure antifreeze to bring the concentration level up to 50 ?Jonas performed an experiment for his science fair project. He learned that rinsing lettuce in vinegar kills more bacteria than rinsing with water or with a popular commercial product. As a follow-up to his project, he wants to determine the percentage of bacteria killed by rinsing with a diluted solution of vinegar. a. How much water and how much vinegar should be mixed to produce 10 cups of a mixture that is 40 vinegar? b. How much pure vinegar and how much 40 vinegar solution should be mixed to produce 10 cups of a mixture that is 60 vinegar?Michelle borrows a total of $5000 in student loans from two lenders. One charges 4.6 simple interest and the other charges 6.2 simple interest. She is not required to pay off the principal or interest for 3yr . However, at the end of 3yr , she will owe a total of $762 for the interest from both loans. How much did she borrow from each lender?Juan borrows $100,000 to pay for medical school. He borrows part of the money from the school whereby he will pay 4.5 simple interest. He borrows the rest of the money through a government loan that will charge him 6 interest. In both cases, he is not required to pay off the principal or interest during his 4yr of medical school. However, at the end of 4yr , he will owe a total of $19,200 for the interest from both loans. How much did he borrow from each source?Stuart pays back two student loans over a 4-yr period. One loan charges the equivalent of 3% simple interest and the other charges the equivalent of 5.5 simple interest. If the total amount borrowed was $24,000 and the total amount of interest paid after 4yr is $3280, find the amount borrowed from each loan.A total of 6000 is invested for 5yr with a total return of $1080 . Part of the money is invested in a fund that returns the equivalent of 2 simple interest. The rest of the money is invested at 4 simple interest. Determine the amount invested in each account.Monique and Tara each make an ice cream sundae. Monique gets 2 scoops of Cherry ice cream and 1 scoop of Mint Chocolate Chunk ice cream for a total of 43g of fat. Tara has 1 scoop of Cherry and 2 scoops of Mint Chocolate Chunk for a total of 47g of fat. How many grams of fat does 1 scoop of each type of ice cream have?Bryan and Jadyn had barbeque potato chips and soda at a football party. Bryan ate 3 oz of chips and drank 2 cups of soda for a total of 700mg of sodium. Jadyn ate 1 oz of chips and drank 3 cups of soda for a total of 350mg of sodium. How much sodium is in 1 oz of chips and how much is in 1 cup of soda?The average weekly salary of two employees is $1350 . One makes $300 more than the other. Find their salaries.The average of an electrician's hourly wage and a plumber's hourly wage is $33 . One day a contractor hires the electrician for 8hr of work and the plumber for 5hr of work and pays a total of $438 in wages. Find the hourly wage for the electrician and for the plumber.A moving sidewalk in an airport moves people between gates. It takes Jason's 9-year -old daughter Josie 40sec to travel 200ft walking with the sidewalk. It takes her 30sec to walk 90ft against the moving sidewalk (in the opposite direction).Find the speed of the sidewalk and find Josie's speed walking on non-moving ground. (See Example 8)A fishing boat travels along the east coast of the United States and encounters the Gulf Stream current. It travels 44mi north with the current in 2hr. It travels 56mi south against the current in 4hr . Find the speed of the current and the speed of the boat in still water.Two runners begin at the same point on a 390-m circular track and run at different speeds. If they run in opposite directions, they pass each other in 30sec . If they run in the same direction, they meet each other in 130sec . Find the speed of each runner.Two particles begin at the same point and move at different speeds along a circular path of circumference 280ft . Moving in opposite directions, they pass in 10sec . Moving in the same direction, they pass in 70sec . Find the speed of each particle.A cleaning company charges $100 for each office it cleans. The fixed monthly cost of $480 for the company includes telephone service and the depreciation on cleaning equipment and a van. The variable cost is $52 per office and includes labor, gasoline, and cleaning supplies. (See Example 9) a. Write a linear cost function representing the cost Cxin$ to the company to clean x offices per month. b. Write a linear revenue function representing the revenue Rxin$ for cleaning x offices per month. c. Determine the number of offices to be cleaned per month for the company to break even. d. If 28 offices are cleaned, will the company make money or lose money?A vendor at a carnival sells cotton candy and caramel apples for $2.00 each. The vendor is charged $100 to set up his booth. Furthermore, the vendor's average cost for each product he produces is approximately $0.75 . a. Write a linear cost function representing the cost Cxin$ to the vendor to produce x products. b. Write a linear revenue function representing the revenue Rxin$ for selling x products. c. Determine the number of products to be produced and sold for the vendor to break even. d. If 60 products are sold, will the vendor make money or lose money?For Exercises 69-70, refer to Figure 8-1 and the narrative at the beginning of this section. Suppose that the price pin$ of theater tickets is influenced by the number of tickets x offered by the theater and demanded by consumers. Supply:p=0.025xDemand:p=0.04x+104 a. Solve the system of equations defined by the supply and demand models. b. What is the equilibrium price? c. What is the equilibrium quantity?For Exercises 69-70, refer to Figure 8-1 and the narrative at the beginning of this section. The price pin$ of a cookbook is determined by the number of cookbooks x demanded by consumers and supplied by the publisher. Supply:p=0.002xDemand:p=0.005x+70 a. Solve the system of equations defined by the supply and demand models. b. What is the equilibrium price? c. What is the equilibrium quantity?a. Sketch the lines defined by y=2xandy=12x+5. b. Find the area of the triangle bounded by the lines in part (a) and the x-axis.a. Sketch the lines defined by y=x+2andy=12x+2. b. Find the area of the triangle bounded by the lines in part (a) and the x-axis.The centroid of a region is the geometric center. For the region shown, the centroid is the point of intersection of the diagonals of the parallelogram. a. Find an equation of the line through the points 3,1 and 7,7. b. Find an equation of the line through the points 0,5 and 4,1. c. Find the centroid of the region.The centroid of the region shown is the point of intersection of the diagonals of the parallelogram. a. Find an equation of the line through the points 2,2 and 3,6. b. Find an equation of the line through the points 1,0 and 0,4. c. Find the centroid of the region.Two angles are complementary. The measure of one angle is 6 less than twice the measure of the other angle. Find the measure of each angle.76PEFor Exercises 77-78, find the measure of angles x and y.For Exercises 77-78, find the measure of angles x and y.Write a system of linear equations with solution set 3,5.Write a system of linear equations with solution set 4,3.Find C and D so that the solution set to the system is 4,1 . Cx+5y=132x+Dy=5Find A and B so that the solution set to the system is 5,2 . 3x+Ay=3Bxy=12Given f(x)=mx+b, find mandbiff3=3andf12=8.Given gx=mx+b, find m and b if g2=1andg4=10.For Exercises 85-86, use the substitution u=1x and v=1y to rewrite the equations in the system in terms of the variables u and v . Solve the system in terms of uandv. Then back substitute to determine the solution set to the original system in terms of xandy. 1x+2y=11x+4y=7For Exercises 85-86, use the substitution u=1x and v=1y to rewrite the equations in the system in terms of the variables u and v . Solve the system in terms of uandv. Then back substitute to determine the solution set to the original system in terms of xandy. 3x+4y=111x2y=5During a race, Marta bicycled 12 mi and ran 4 mi in a total of 1hr20min43hr. In another race, she bicycled 21 mi and ran 3 mi in 1hr40min53hr. Determine the speed at which she bicycles and the speed at which she runs. Assume that her bicycling speed was the same in each race and that her running speed was the same in each race.Shelia swam 1mi and ran 6mi in a total of 1hr15min54hr. In another training session she swam 2mi and ran 8mi in a total of 2hr . Determine the speed at which she swims and the speed at which she runs. Assume that her swimming speed was the same each day and that her running speed was the same each day.A certain pickup truck gets 16 mpg in the city and 22 mpg on the highway. If a driver drives 254mi on 14gal of gas, determine the number of city miles and highway miles that the truck was driven.A sedan gets 12 mpg in the city and 18 mpg on the highway. If a driver drives a total of 420mi on 26gal of gas, how many miles in the city and how many miles on the highway did he drive?A system of linear equations in x and y can represent two intersecting lines, two parallel lines, or a single line. Describe the solution set to the system in each case.When solving a system of linear equations in two variables using the substitution or addition method, explain how you can detect whether the equations are dependent.When solving a system of linear equations in two variables using the substitution or addition method, explain how you can detect whether the system is inconsistent.Consider a system of linear equations in two variables in which the solution set is (x,x+2)x is any real number}. Why do we say that the equations in the system are dependent?A 50-lb weight is supported from two cables and the system is in equilibrium. The magnitudes of the forces on the cables are denoted by F1andF2, respectively. An engineering student knows that the horizontal components of the two forces (shown in red) must be equal in magnitude. Furthermore, the sum of the magnitudes of the vertical components of the forces (shown in blue) must be equal to 50-lb to offset the downward force of the weight. Find the values of F1andF2. Write the answers in exact form with no radical in the denominator. Also give approximations to 1 decimal place.For Exercises 96-99, use a graphing utility to approximate the solution to the system of equations. Round the x and y values to 3 decimal places. y=3.729x+6.958y=2.615x8.71397PEFor Exercises 96-99, use a graphing utility to approximate the solution to the system of equations. Round the x and y values to 3 decimal places. 0.25x+0.04y=0.426.755x+2.5y=38.1For Exercises 96-99, use a graphing utility to approximate the solution to the system of equations. Round the x and y values to 3 decimal places. 0.36x0.075y=0.8130.066x+0.008y=0.194Determine if the ordered triple is a solution to the system. 5xy+3z=73x+4yz=59x+5y+7z=1 a.2,6,3b.1,2,0Solve the system. 2xy+5z=7x+4y2z=13x+2y+z=73SPRepeat Example 4 with the given system. x+y+4z=13x+y4z=34xy+8z=25SPNicolas mixes three solutions of acid with concentrations of 10,15, and 5. He wants to make 30L of a mixture that is 12 acid and he uses four times as much of the 15 solution as the 5 solution. How much of each of the three solutions must he use?7SPThe graph of a linear equation in two variables is a line in a two-dimensional coordinate system. The graph of a linear equation in three variables is a in a three-dimensional coordinate system.A solution to a system of linear equations in three variables is an ordered that satisfies each equation in the system. Graphically, this is a point of of three planes.3PEFor Exercises 3-4, find three ordered triples that are solutions to the linear equation in three variables. 3x5y+z=15For Exercises 5-8, determine if the ordered triple is a solution to the system of equations. (See Example 1) x+3y7z=72x+4y+z=163x5y+6z=9 a.2,3,0b.2,4,1For Exercises 5-8, determine if the ordered triple is a solution to the system of equations. (See Example 1) 2x+3y+z=12x+y2z=93x+2yz=7 a. 2,5,1 b. 1,4,2For Exercises 5-8, determine if the ordered triple is a solution to the system of equations. (See Example 1) x+y+z=2x+2yz=23x+5yz=6 a. 2,0,0 b. 1,2,1For Exercises 5-8, determine if the ordered triple is a solution to the system of equations. (See Example 1) xy+z=33x+4yz=15x+7yz=1 a. 1,2,6 b. 3,1,5For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) x2y+z=93x+4y+5z=92x+3yz=12For Exercise 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solution to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 2xy+z=6x+5yz=103x+y3z=12For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 4x=3y2z52x+y=y+z66xy+z=x5y8For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 3x=5yz+13xyz=x35x+y=3y3z4For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 2x+5z=23y7z=95x+9y=22For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 3x2y=85y+6z=27x+11z=33For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 4x3y=03y+z=14xz=12For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 4xy+2z=13x+5yz=29x15y+3z=0For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 2x=3y6z16y=12z10x+93z=6y3x1For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 5x=2y3z34y=110x5z2z=5x6y19PEFor Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 0.2x=0.1y0.6z0.004x+0.005y0.001z=030x=50z20y21PE22PE23PEFor Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) x+72y+12z=434x+y+12z=1110x25y310z=125PE26PE27PE28PEFor Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 3x+y=64z+y4=6y+5z3x+4y+z=0For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 4xy=8zy3=3x+4zx+3y+3z=1For Exercises 9-32, solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. (See Examples 2-5) 3x+4yz=4x+2y+z=412x+16y4z=1632PE33PEFor Exercises 33-36, solve the system from the indicated exercise and write the general solution. (See Example 5) Exercise 2035PEFor Exercises 33-36, solve the system from the indicated exercise and write the general solution. (See Example 5) Exercise 32For Exercises 37-38, the general solution is given for a system of linear equations. Find three individual solutions to the system. x+4y+2z=4x3yz=2x+yz=2 Solution: 2z+4,z+2,zzisanyrealnumberFor Exercises 37-38, the general solution is given for a system of linear equations. Find three individual solutions to the system. 2x3y+z=1x+4yz=35x2y+z=5 Solution: x3x+4,11x+13xisanyrealnumberDevon invested $8000 in three different mutual funds. A fund containing large cap stocks made 6.2 return in 1yr . A real estate fund lost 13.5 in 1yr, and a bond fund made 4.4 in 1yr. The amount invested in the large cap stock fund was twice the amount invested in the real estate fund. If Devon had a net return of $66 across all investments, how much did he invest in each fund? (See Example 6)Pierre inherited $120,000 from his uncle and decided to invest the money. He put part of the money in a money market account that earns 2.2 simple interest. The remaining money was invested in a stock that returned 6 in the first year and a mutual fund that lost 2 in the first year. He invested $10,000 more in the stock than in the mutual fund, and his net gain for 1yr was $2820. Determine the amount invested in each account.A basketball player scored 26 points in one game. In basketball, some baskets are worth 3 points, some are worth 2 points, and free-throws are worth 1 point He scored four more 2-point baskets than he did 3 -point baskets. The number of free-throws equaled the sum of the number of 2-pointand3-point shots made. How many free-throws, 2-point shots, and 3-point shots did he make?42PE43PEA theater charges $50 per ticket for seats in Section A, $30 per ticket for seats in Section B, and $20 per ticket for seats in Section C. For one play, 4000 tickets were sold for a total of $120,000 in revenue. If 1000 more tickets in Section B were sold than the other two sections combined, how many tickets in each section were sold?45PEA package in the shape of a rectangular solid is to be mailed. The combination of the girth (perimeter of a cross section defined by w and h ) and the length of the package is 48in . The width is 2in . greater than the height, and the length is 12in . greater than the width. Find the dimensions of the package.The measure of the largest angle in a triangle is 1008 larger than the sum of the measures of the other two angles. The measure of the smallest angle is two-thirds the measure of the middle angle. Find the measure of each angle.48PEa. Show that the points 1,0,3,10, and 2,15 are not collinear by finding the slope between 1,0 and 3,10 , and the slope between 3,10 and 2,15. (See Example 7) b. Find an equation of the form y=ax2+bx+c that defines the parabola through the points. c. Use a graphing utility to verify that the graph of the equation in part (b) passes through the given points.a. Show that the points 2,9,1,6, and 4,3 are not collinear by finding the slope between 2,9 and 1,6 , and the slope between 2,9 and 4,3. b. Find an equation of the form y=ax2+bx+c that defines the parabola through the points. c. Use a graphing utility to verify that the graph of the equation in part (b) passes through the given points.51PEFor Exercises 51-52, find an equation of the form y=ax2+bx+c that defines the parabola through the three noncollinear points given. 0,4,2,6,3,31The motion of an object traveling along a straight path is given by st=12at2+v0t+s0, where st is the position relative to the origin at time t. For Exercises 53-54, three observed data points are given. Find the values of a,v0, and s0 . s1=30,s2=54,s3=8254PEMany statistics courses cover a topic called multiple regression. This provides a means to predict the value of a dependent variable y based on two or more independent variables x1,x2,...,xn. The model y=ax1+bx2+c is a linear model that predicts y based on two independent variables x1 and x2. While statistical techniques may be used to find the values of a,b, and c based on a large number of data points, we can form a crude model given three data values x1,x2,y . Use the information given in Exercises 55-56 to form a system of three equations and three variables to solve for a,b, and c. The selling price of a homey (in $1000 ) is given based on the living area x1 (in 100ft2 ) and on the lot size x2 (in acres). a. Use the data to create a model of the form y=ax1+bx2+c . b. Use the model from part (a) to predict the selling price of a home that is 2000ft2 on a 0.4-acre lot.Many statistics courses cover a topic called multiple regression. This provides a means to predict the value of a dependent variable y based on two or more independent variables x1,x2,...,xn. The model y=ax1+bx2+c is a linear model that predicts y based on two independent variables x1 and x2. While statistical techniques may be used to find the values of a,b, and c based on a large number of data points, we can form a crude model given three data values x1,x2,y . Use the information given in Exercises 55-56 to form a system of three equations and three variables to solve for a,b, and c. The gas mileage y (in mpg) for city driving is given based on the weight of the vehicle x1 (in lb) and on the number of cylinders. a. Use the data create a model of the form y=ax1+bx2+c. b. Use the model from part (a) to predict the gas mileage of a vehicle that is 3800lb and has 6 cylinders.Give a geometric description of the solution set to a linear equation in three variables.If a system of linear equations in three variables has no solution, then what can be said about the three planes represented by the equations in the system?Explain the procedure presented in this section to solve a system of linear equations in three variables.Explain how to check a solution to a system of linear equations in three variables.For Exercises 61-62, find all solutions of the form a,b,c,d. 2a+bc+d=73b+2c2d=11a+3c+3d=144a+2b5c=662PE63PE64PE65PE66PEFor Exercises 67-68, find the constants A and B so that the two polynomials are equal. (Hint: Create a system of linear equations by equating the constant terms and by equating the coefficients on the x terms and x2 terms.) 11x2+26x5=2Ax2+5Ax+3A+Bx22Bx8B+2Cx27Cx4CFor Exercises 67-68, find the constants A and B so that the two polynomials are equal. (Hint: Create a system of linear equations by equating the constant terms and by equating the coefficients on the x terms and x2 terms.) 3x2+37x82=Ax2+Ax12A+3Bx210Bx+3B+3Cx2+11Cx4CSet up the form for the partial fraction decomposition for the given rational expressions. a. x+183x+1x+4 b. x2+3x+8x3+4x2+4x2SP3SP4SP5SP6SPThe process of decomposing a rational expression into two or more simpler fractions is called partial .2PEWhen setting up a partial fraction decomposition, if the denominator of a fraction is a quadratic polynomial irreducible over the integers, then the numerator should be (constant/linear). That is, should the numerator be set up as A or Ax+B?In what situation should long division be used before attempting to decompose a rational expression into partial fractions?For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) x37x+42x3For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 20x4x53x+1For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 8x10x22xFor Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) y12y2+3yFor Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 6w7w2+w6For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 10t11t2+5t6For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) x2+26x+100x3+10x2+25xFor Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 3x2+2x+8x3+4x2+4x13PEFor Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 17x27x+187x3+42xFor Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 2x3x2+13x5x4+10x2+25For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 3x34x2+11x12x4+6x2+9For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 5x24x+8x4x2+x+4For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) x2+15x6x+6x2+2x+6For Exercises 5-20, set up the form for the partial fraction decomposition. Do not solve for A,B,C, and so on. (See Examples 1-2) 2x5+3x3+4x2+5xx+23x2+2x+7220PEFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) x37x+42x3For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 20x4x53x+1For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 8x10x22xFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) y12y2+3yFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 6w7w2+w6For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 10t11t2+5t6For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) x2+26x+100x3+10x2+25xFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 3x2+2x+8x3+4x2+4xFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 13x2+2x+452x3+18xFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 17x27x+187x3+42xFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) x43x3+13x228x+28x3+7xFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) x44x3+11x213x+12x3+2xFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 2x3x2+13x5x4+10x2+25For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 3x34x2+11x12x4+6x2+9For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 5x24x+8x4x2+x+436PE37PEFor Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 3x34x2+6x7x4+5x2+4For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 2x311x24x+24x23x10For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 3x2+11x2+x+10x2+3x4For Exercises 21-42, find the partial fraction decomposition. (See Examples 3-6) 3x2+2x2x5x2+2x+142PEa. Factor. x3x221x+45 (Hint: Use the rational zero theorem.) b. Find the partial fraction decomposition for 3x2+35x70x3x221x+45a. Factor, x32x27x+4 (Hint: Use the rational zero theorem.) b. Find the partial fraction decomposition for 10x2+17x17x3+2x27x+4a. Factor, x3+6x2+12x+8 (Hint: Use the rational zero theorem.) b. Find the partial fraction decomposition for 3x2+8x5x3+6x2+12x+8.a. Factor, x39x2+27x27 (Hint: Use the rational zero theorem.) b. Find the partial fraction decomposition for 2x217x+37x39x2+27x27 .Write an informal explanation of partial fraction decomposition.Suppose that a proper rational expression has a single repeated linear factor ax+b3 in the denominator. Explain how to set up the partial fraction decomposition.What is meant by a proper rational expression?Given an improper rational expression, what must be done first before the technique of partial fraction decomposition may be performed?a. Determine the partial fraction decomposition for 2nn+2 . b. Use the partial fraction decomposition for 2nn+2 to rewrite the infinite sum 213+224+235+246+257 c. Determine the value of 1n+2 as n. d. Find the value of the sum from part (b).a. Determine the partial fraction decomposition for 3nn+3. b. Use the partial fraction decomposition for 3nn+3 to rewrite the infinite sum 314+325+336+347+358 c. Determine the value of 1n+3 as n. d. Find the value of the sum from part (b).For Exercises 53-54, find the partial fraction decomposition. Assume that a and b are nonzero constants. 1xa+bxFor Exercises 53-54, find the partial fraction decomposition. Assume that a and b are nonzero constants. 1a2x2For Exercises 55-56, find the partial fraction decomposition for the given expression. [Hint. Use the substitution u=ex and recall that e2x=ex2.] 5ex+7e2x+3ex+2For Exercises 55-56, find the partial fraction decomposition for the given expression. [Hint. Use the substitution u=ex and recall that e2x=ex2.] 3ex22e2x+3ex4Solve the system by using the substitution method. 2x+y=5x2+y2=50Solve the system by using the substitution method. x2+y2=90y=xSolve the system by using the addition method. x2+y2=17x22y2=31Solve the system by using the addition method. x2+y2=164x2+9y2=36The perimeter of a rectangular rug is 40ft and the area is 96ft2. Find the dimensions of the rug.A system of equations in two variables is a system in which one or more equations in the system is nonlinear.A solution to a nonlinear system of equations in two variables is an pair with real valued coordinates that satisfies each equation in the system. Graphically, a solution is a point of of the graphs of the equations.For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) y=x222xy=2For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) y=x2+3y2x=0For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) x2+y2=25x+y=1For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) x2+y2=253y=4xFor Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) y=xx2+y2=20For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) x2+y2=10y=x2For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) x+22+y2=9y=2x4For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) x2+y32=4y=x4For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) y=x3y=xFor Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) y=x3y=xFor Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) y=x22+5y=2x+1For Exercises 3-14, a. Graph the equations in the system. b. Solve the system by using the substitution method. (See Examples 1-2) y=x+321y=2x+5For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) 2x2+3y2=11x2+4y2=8For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) 3x2+y2=214x22y2=2For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) x2xy=202x2+3xy=44For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) 4xy+3y2=92xy+y2=5For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) 5x22y2=12x23y2=4For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) 6x2+5y2=387x23y2=9For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) x2=1y29x24y2=36For Exercises 15-22, solve the system by using the addition method. (See Examples 3-4) 4x2=4y216y2=144+9x2For Exercises 23-34, solve the system by using any method. x24xy+4y2=1x+y=4For Exercises 23-34, solve the system by using any method. x26xy+9y2=0xy=2For Exercises 23-34, solve the system by using any method. y=x2+4x+5y=4x+5For Exercises 23-34, solve the system by using any method. y=x26x+9y=2x+5For Exercises 23-34, solve the system by using any method. y=x2y=1xFor Exercises 23-34, solve the system by using any method. y=1xy=xFor Exercises 23-34, solve the system by using any method. x2+y42=25y=x2+9For Exercises 23-34, solve the system by using any method. x102+y2=100x=y2For Exercises 23-34, solve the system by using any method. y=x2+6x7y=x210x+23For Exercises 23-34, solve the system by using any method. y=x2+6x9y=x22x3For Exercises 23-34, solve the system by using any method. x24+y216=1x2+y=4For Exercises 23-34, solve the system by using any method. x24+y2=1x=2y2+2For Exercises 35-36, use the substitutions u=1x2 and v=1y2 to solve the system of equations. 4x23y2=235x2+1y2=14For Exercises 35-36, use the substitutions u=1x2 and v=1y2 to solve the system of equations. 3x2+1y2=135x21y2=5Find two numbers whose sum is 12 and whose product is 35 .Find two numbers whose sum is 9 and whose product is 36 .The sum of the squares of two positive numbers is 29 and the difference of the squares of the numbers is 21 . Find the numbers.The sum of the squares of two negative numbers is 145 and the difference of the squares of the numbers is 17 . Find the numbers.The difference of two positive numbers is 2 and the difference of their squares is 44 . Find the numbers.The sum of two numbers is 4 and the difference of their squares is 64. Find the numbers.The ratio of two numbers is 3 to 4 and the sum of their squares is 225 . Find the numbers.The ratio of two numbers is 5 to 12 and the sum of their squares is 676 . Find the numbers.Find the dimensions of a rectangle whose perimeter is 36m and whose area is 80m2.Find the dimensions of a rectangle whose perimeter is 56cm and whose area is 192cm2.The floor of a rectangular bedroom requires 240ft2 of carpeting. Molding is placed around the base of the floor except at two 3-ft doorways. If 58ft of molding is required around the base of the floor, determine the dimensions of the floor. (See Example 5)An electronic sign for a grocery store is in the shape of a rectangle. The perimeter of the sign is 72ft and the area is 320ft2. Find the length and width of the sign.A rental truck has a cargo capacity of 288ft3. 10-ft pipe just fits resting diagonally on the floor of the truck. If the cargo space is 6ft high, find the dimensions of the truck.A rectangular window has a 15-yd diagonal and an area of 108yd2. Find the dimensions of the window.An aquarium is 16 in. high with volume of 4608in.3 (approximately 20gal ). If the amount of glass used for the bottom and four sides is 1440in.2 , determine the dimensions of the aquarium.52PEThe hypotenuse of a right triangle is 65ft. The sum of the lengths of the legs is 11ft. Find the lengths of the legs.The hypotenuse of a right triangle is 73 in. The sum of the lengths of the legs is 11in. Find the lengths of the legs.A ball is kicked off the side of a hill at an angle of elevation of 30. The hill slopes downward 30 from the horizontal. Consider a coordinate system in which the origin is the point on the edge of the hill from which the ball is kicked. The path of the ball and the line of declination of the hill can be approximated by y=x2192+33xPathofthebally=33xLineofdeclinationofthehill Solve the system to determine where the ball will hit the ground.A child kicks a rock off the side of a hill at an angle of elevation of 60. The hill slopes downward 30 from the horizontal. Consider a coordinate system in which the origin is the point on the edge of the hill from which the rock is kicked. The path of the rock and the line of declination of the hill can be approximated by y=x236+3xPathoftherocky=33xLineofdeclinationofthehill Solve the system to determine where the rock will hit the ground.What is the difference between a system of linear equations and a system of nonlinear equations?Describe a situation in which the addition method is an efficient technique to solve a system of nonlinear equations.The attending physician in an emergency room treats an unconscious patient suspected of a drug overdose. The physician does not know the initial concentration A0 of the drug in the bloodstream at the time of injection. However, the physician knows that after 3hr , the drug concentration in the blood is 0.69g/dL and after 4hr, the concentration is 0.655g/dL. The model At=A0ekt represents the drug concentration Ating/dL in the bloodstream t hours after injection. The value of k is a constant related to the rate at which the drug is removed by the body. a. Substitute 0.69 for At and 3 for t in the model and write the resulting equation. b. Substitute 0.655 for At and 4 for t in the model and write the resulting equation. c. Use the system of equations from parts (a) and (b) to solve for k . Round to 3 decimal places. d. Use the system of equations from parts (a) and (b) to approximate the initial concentration A0 ing/dL at the time of injection. Round to 2 decimal places. e. Determine the concentration of the drug after 12hr. Round to 2 decimal places.A patient undergoing a heart scan is given a sample of fluorine- 1818F. After 4hr , the radioactivity level in the patient is 44.1MBq (megabecquerel). After 5hr, the radioactivity level drops to 30.2MBq. The radioactivity level Qt can be approximated by Qt=Q0ekt, where t is the time in hours after the initial dose Q0 is administered. a. Determine the value of k . Round to 4 decimal places. b. Determine the initial dose, Q0 . Round to the nearest whole unit. c. Determine the radioactivity level after 12hr. Round to 1 decimal place.The population Pt of a culture of bacteria grows exponentially for the first 72hr according to the model Pt=P0ekt. The variable t is the time in hours since the culture is started. The population of bacteria is 60,000 after 7hr. The population grows to 80,000 after 12hr. a. Determine the constant k to 3 decimal places. b. Determine the original population P0 . Round to the nearest thousand. c. Determine the time required for the population to reach 300,000 . Round to the nearest hour.An investment grows exponentially under continuous compounding. After 2yr , the amount in thee account is $7328.70. After 5yr, the amount in the account is $8774.10. Use the model At=Pert to a. Find the interest rate r. Round to the nearest percent. b. Find the original principal P. Round the nearest dollar. c. Determine the amount of time required for the account to reach a value of $15,000. Round to the nearest year.