Chapter 8.3, Problem 33E

To determine

For the graph of $y=ax+b$ where $a\ne 0$ , show that the x -intercept is $-\frac{b}{a}$ and the y -intercept is b . Use the results to find the intercepts of the graph of given equation.

Answer to Problem 33E

The x -intercept of our given equation would be $(-4,0)$ .

The y -intercept of our given equation would be $(0,12)$ .

Explanation of Solution

Given:

An equation: $y=3x+12$

Calculation:

We know that to find x -intercept, we need to substitute $y=0$ in our given equation and solve for x .

Let us solve for x -intercept as shown below:

  $\begin{array}{l}y=ax+b\left(\text{Given equation}\right)\\ 0=ax+b\left(\text{Substituting }y=0\right)\\ 0-b=ax+b-b\left(\text{Subtracting }b\text{ from both sides}\right)\\ -b=ax\\ ax=-b\left(\text{Switching sides}\right)\\ \frac{ax}{a}=-\frac{b}{a}\left(\text{Dividing both sides by }a\right)\\ x=-\frac{b}{a}\end{array}$

Hence proved.

To find y- intercept, we will substitute $x=0$ in our given equation and solve for yas shown below:

  $\begin{array}{l}y=ax+b\left(\text{Given equation}\right)\\ y=a(0)+b\left(\text{Substituting }x=0\right)\\ y=0+b\\ y=b\end{array}$

Hence proved.

Upon comparing our given equation $y=3x+12$ with $y=ax+b$ , we can see that the value of a is 3 and the value of b is 12. Therefore, the y -intercept of our given equation is $(0,12)$ .

  $\begin{array}{l}x=-\frac{b}{a}\\ x=-\frac{12}{3}\\ x=-4\end{array}$

Therefore, the x -intercept of our given equation is $(-4,0)$ .