Chapter 11.1, Problem 82AYU

a.

To determine

To construct: The system of linear equations with three variables such that it has no solution.

Answer to Problem 82AYU

The system of linear equations with three variables such that it has no solution is, $\{\begin{array}{c}x+y+2z=12\\ 3x+3y+6z=16\\ 10x+y+3z=9\end{array}$ .

Explanation of Solution

Given information:

The system of linear equations with three variables such that it has no solution.

Consider the system of linear equations with three variables.

Recall that parallel lines have no solution because they never intersect.

Two lines are said to be parallel if the ratio of the coefficients of the lines are equal.

Suppose take an equation of line with three variables as, $x+y+2z=12$ .

Now, multiply the coefficient of equation by three to obtain the second equation $3x+3y+6z=16$ .

Take a random third equation $10x+y+3z=9$

Now, system of equations so formed is,

  $\{\begin{array}{c}x+y+2z=12\\ 3x+3y+6z=16\\ 10x+y+3z=9\end{array}$

Since, first two equations are parallel because the coefficients have same ratio so the system has no solution. Because two lines out of three are parallel.

b.

To determine

To construct: The system of linear equations with three variables such that it has exactly one solution.

Answer to Problem 82AYU

The system of linear equations with three variables such that it has exactly one solution is, $\{\begin{array}{c}x+y+2z=12\\ 5x+2y+9z=10\\ 4x+16y+3z=9\end{array}$ .

Explanation of Solution

Given information:

The system of linear equations with three variables such that it has exactly one solution.

Consider the system of linear equations with three variables.

Recall that intersecting lines can either intersect at one or two points.

Two lines are said to be intersecting if the ratio of the coefficients of the lines are not equal

Suppose take an equation of line with three variables as, $x+y+2z=12$ .

Now, the second equation $5x+2y+9z=10$ .

Take a random third equation $4x+16y+3z=9$

Now, system of equations so formed is,

  $\{\begin{array}{c}x+y+2z=12\\ 5x+2y+9z=10\\ 4x+16y+3z=9\end{array}$

Since, the coefficients of all the three lines are in same ratio so the system has exactly one solution.

c.

To determine

To construct: The system of linear equations with three variables such that it has infinitely many solutions.

Answer to Problem 82AYU

The system of linear equations with three variables such that it has infinitely many solutions is, $\{\begin{array}{c}x+y+2z=12\\ 2x+2y+4z=24\\ x+3y+6z=36\end{array}$ .

Explanation of Solution

Given information:

The system of linear equations with three variables such that it has infinitely many solutions.

Consider the system of linear equations with three variables.

Recall that coincident lines have infinitely many solutions.

Two lines are said to be coincident if the ratio of the coefficients of the lines are equal.

Suppose take an equation of line with three variables as, $x+y+2z=12$ .

Now, the second equation $2x+2y+4z=24$ , that is multiply the first equation by 2.

Take a random third equation $x+3y+6z=36$

Now, system of equations so formed is,

  $\{\begin{array}{c}x+y+2z=12\\ 2x+2y+4z=24\\ x+3y+6z=36\end{array}$

Since, the coefficients of all the two lines are in same ratio so the system has infinitely many solutions.