Chapter 10, Problem 2CA

a.

To determine

To fill: The blank space for m and b so that the given statement is true.

Answer to Problem 2CA

When m = 1 and b = 5, the graph of the equation passes through the point $\left(-1,4\right)$ .

Explanation of Solution

Given information:

The equation: $y=mx+b$

The statement:

When m = ___ and b = ___, the graph of the equation passes through the point $\left(-1,4\right)$ .

The given equation is $y=mx+b$ .

Consider that the slope, m be 1.

Substitute 1 for m , $-1$ for x and 4 for y in the given equation.

  $\begin{array}{c}4=1\left(-1\right)+b\\ b-1=4\\ b=4+1\\ b=5\end{array}$

Hence, when m = 1 and b = 5, the graph of the equation passes through the point $\left(-1,4\right)$ .

b.

To determine

To fill: The blank space for m and b so that the given statement is true.

Answer to Problem 2CA

When m = 1 and b = −3, the graph of the equation has a positive slope and passes through the point $\left(-2,-5\right)$ .

Explanation of Solution

Given information:

The equation: $y=mx+b$

The statement:

When m = ___ and b = ___, the graph of the equation has a positive slope and passes through the point $\left(-2,-5\right)$ .

The given equation is $y=mx+b$ .

Consider that the slope, m be 1.

Substitute 1 for m , $-2$ for x and $-5$ for y in the given equation.

  $\begin{array}{c}-5=1\left(-2\right)+b\\ b-2=-5\\ b=-5+2\\ b=-3\end{array}$

Hence, when m = 1 and b = −3, the graph of the equation has a positive slope and passes through the point $\left(-2,-5\right)$ .

c.

To determine

To fill: The blank space for m and b so that the given statement is true.

Answer to Problem 2CA

When m = $-\frac{1}{4}$ and b = $\frac{25}{4}$ , the graph of the equation is perpendicular to the graph of $y=4x-3$ and passes through the point $\left(1,6\right)$ .

Explanation of Solution

Given information:

The equation: $y=mx+b$

The statement:

When m = ___ and b = ___, the graph of the equation is perpendicular to the graph of $y=4x-3$ and passes through the point $\left(1,6\right)$ .

The given equation is $y=mx+b$ .

Since, the lines are perpendicular so the slopes of the lines are negative reciprocal of each other.

So, the slope is $m=-\frac{1}{4}$ .

Substitute $-\frac{1}{4}$ for m , 1 for x and 6 for y in the given equation.

  $\begin{array}{c}6=\left(-\frac{1}{4}\right)\left(1\right)+b\\ b-\frac{1}{4}=6\\ b=6+\frac{1}{4}\\ b=\frac{25}{4}\end{array}$

Hence, when m = $-\frac{1}{4}$ and b = $\frac{25}{4}$ , the graph of the equation is perpendicular to the graph of $y=4x-3$ and passes through the point $\left(1,6\right)$ .