For Exercises 1-4, solve the system of equations using
- The substitution method or the addition method (see Sections 9. 1 and 9.2).
- Gaussian elimination (see Section 10.1)
- Gauss-Jordan elimination (see Section 10.1).
- The inverse of the coefficient matrix (see Section 10.4).
- Cramer's rule (see Section 10.5).
1.
a.
To solve: The system of equations using substitution method.
Answer to Problem 1PRE
The required solution of the system of equations by substitution method is
Explanation of Solution
Given:
The given system of equationsis
Method Used:
The steps used in substitution method are:
Step1: Choose one of the two equations and solve it for one of the two variables. (Make sure avoiding fractions, if possible.)
Step2: Substitute the value of variable of step 1 into the equation that is not used in step 1 and then solve resulted linear equation for one variable.
Step3: Substitute the result of step 2 into the expression obtained in step 1 to find the value of the other variable.
Calculations:
The value of variable x from equation (1) is
Now substitute value of y back into the first equation
Thus
Conclusion:
The solution of given system of equations by substitution method is
b.
To solve: The system of equations using Gaussian elimination method.
Answer to Problem 1PRE
The solution of the system of equations using Gaussian elimination method is
Explanation of Solution
Given:
The system of equations in part (a)
Method used:
The steps used in Gauss elimination are:
Step 1: Write augmented matrix for the system of equations
Step 2: Using elementary operations write augmented matrix in “row echelon form”
Step 3: Using back substitution solve the resulted set of equations
Calculations:
The given system of equations can be written as:
The coefficient matrix and the augmented matrix for the given system of equations are
Thus, value of
The solution of system of equations is
Conclusion:The solution of the system of equations using Gaussian elimination method is
c.
To solve: The system of equations using Gauss- Jordan elimination method.
Answer to Problem 1PRE
The solution of the system of equations using Gauss- Jordan elimination method is
Explanation of Solution
Given:
The system of equations in part (a)
Method used:
In Gauss- Jordan elimination method a “reduced row Echelon matrix” is obtained using appropriate elementary row operations as given below:
Step 1: Choosing the leftmost nonzero column and using row operation get a 1 at the top.
Step2: Use multiples of the rows containing 1 from step 1, and get zeros in all remaining places in the column containing this 1.
Step 3: Repeat step 1 with the sub-matrixformed by deleting (in mind only) the row used in step 2 and all rows above this row.
Step 4: Repeat step 2 with the entire matrix until the entire matrix get transformed in reduced row Echelon form.
Calculations:
The given system of equations can be written as:
The coefficient matrix and the augmented matrix for the given system of equations are
Now applying row operations:
The solution of system of equations is
Conclusion: The solution of system of equations by Gauss-Jordan elimination method is
d.
To solve: The system of equations using inverse of the coefficient matrix.
Answer to Problem 1PRE
The solution of the system of equations using inverse of coefficient matrix is
Explanation of Solution
Given:
The system of equations in part (a)
Method/ Formula used:
The system of equations can be written as
Calculations:
The given system of equations can be written as:
The inverse of A using formula is
Therefore, the solution of the system of equations is
The solution of system of equations is
Conclusion: The solution of system of equations by inverse of coefficient matrix method is
e.
To solve: The system of equations using Cramer’s rule.
Answer to Problem 1PRE
The solution of the system of equations using Cramer’s rule is
Explanation of Solution
Given:
The system of equations in part (a)
Method/ Formula used:
For two variable system of equations
Calculations:
The given system of equations can be written as:
For the system of equations, the determinant of coefficient matrix
Therefore, values of x and y are
The solution of system of equations is
Conclusion: The solution of the system of equations usingCramer’s is
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