Chapter 8.8, Problem 26E

a.

To determine

To find:

The values of m and b for which the given system has exactly one solution.

Answer to Problem 26E

The given system will have exactly one solution, when $m=2\text{}\text{and }b=4$ .

Explanation of Solution

Given:

A system of equations:

  $\begin{array}{l}y=3x-2\\ y=mx+b\end{array}$

Calculation:

We know that a system of equations will have exactly one solution, when the slopes of both lines are different.

So the given system will have exactly one solution, when the value of m is other than 3.

Therefore, given system will have exactly one solution, when $m=2\text{}\text{and }b=4$ .

b.

To determine

To find:

The values of m and b for which the given system has no solution.

Answer to Problem 26E

The given system will have no solution, when $m=3\text{}\text{and }b=4$ .

Explanation of Solution

Given:

A system of equations:

  $\begin{array}{l}y=3x-2\\ y=mx+b\end{array}$

Calculation:

We know that a system of equations will have no solution, when the slopes of both lines are same and the y-intercepts are different.

So the given system will have no solution, when the value of m is 3 and the value of b is other than $-2$ .

Therefore, given system will have no solution, when $m=3\text{}\text{and }b=4$ .

c.

To determine

To find:

The values of m and b for which the given system has infinitely many solutions.

Answer to Problem 26E

The given system will have infinitely many solutions, when $m=3\text{}\text{and }b=-2$ .

Explanation of Solution

Given:

A system of equations:

  $\begin{array}{l}y=3x-2\\ y=mx+b\end{array}$

Calculation:

We know that a system of equations will have infinitely many solutions, when the slope and y-intercepts of both lines are same.

So the given system will have infinitely many solutions, when the value of m is 3 and the value of b is $-2$ .

Therefore, given system will have infinitely many solutions, when $m=3\text{}\text{and }b=-2$ .