Question

1. write an equation for the line that passes through each of the following points with the specified slope. write each equation in point - slope form and then in slope - intercept form, i.e. $y = mx + b$.

(a) slope = - 4
point: $(-3,9)$
(b) slope = $-\frac{2}{3}$
point: $(12,2)$

Explanation:

Part (a)

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.
Given $m=-4$ and $(x_1,y_1)=(-3,9)$, substitute into the point - slope form:
$y - 9=-4(x - (-3))$
Simplify the equation:
$y - 9=-4(x + 3)$

Step2: Convert to slope - intercept form ($y=mx + b$)

Expand the right - hand side:
$y - 9=-4x-12$
Add 9 to both sides of the equation:
$y=-4x-12 + 9$
Simplify the right - hand side:
$y=-4x-3$

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)$ is a point on the line and $m$ is the slope.
Given $m =-\frac{2}{3}$ and $(x_1,y_1)=(12,2)$, substitute into the point - slope form:
$y - 2=-\frac{2}{3}(x - 12)$

Step2: Convert to slope - intercept form ($y=mx + b$)

Expand the right - hand side:
$y - 2=-\frac{2}{3}x+8$
Add 2 to both sides of the equation:
$y=-\frac{2}{3}x+8 + 2$
Simplify the right - hand side:
$y=-\frac{2}{3}x + 10$

Answer:

Point - slope form: $y - 9=-4(x + 3)$; Slope - intercept form: $y=-4x-3$

Part (b)