Question

use the pairs of numbers provided in the table below to answer the following questions:

x	y
12	-4
16	-7

what is the rate of change?
what is the y-value when the x-value is 15?
infinite possibilities impossible

Explanation:

Step1: Calculate rate of change

The rate of change (slope) formula is $m=\frac{y_2-y_1}{x_2-x_1}$. Use first two points: $(8,-1)$ and $(12,-4)$.
$m=\frac{-4-(-1)}{12-8}=\frac{-3}{4}$
Verify with second and third points: $(12,-4)$ and $(16,-7)$
$m=\frac{-7-(-4)}{16-12}=\frac{-3}{4}$

Step2: Find linear equation

Use point-slope form $y-y_1=m(x-x_1)$. Use $(8,-1)$:
$y-(-1)=-\frac{3}{4}(x-8)$
Simplify to slope-intercept form:
$y+1=-\frac{3}{4}x+6$
$y=-\frac{3}{4}x+5$

Step3: Solve for y when x=15

Substitute $x=15$ into the equation:
$y=-\frac{3}{4}(15)+5=-\frac{45}{4}+\frac{20}{4}=-\frac{25}{4}=-6.25$

Answer:

Rate of change: $\boldsymbol{-\frac{3}{4}}$
y-value when x=15: $\boldsymbol{-\frac{25}{4}}$ (or $\boldsymbol{-6.25}$)