Question

try your best and good luck!

1. write an equation of each line with the given information.

a. slope: 4; y-intercept: -3
b. slope: -2; point: (2, -1)

1. write an equation of each line on the graph.
2. write an equation of the line that passes through each pair of points.

a. (-3, -3), (0, 6)
b. (-2, 5), (3, -4)

1. parallel lines have ____ slopes. give an example.
2. perpendicular lines have __ __ slopes. give an example.

Explanation:

Problem 1a:

Step1: Recall slope - intercept form

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step2: Substitute values

Given that $m = 4$ and $b=-3$, substitute these values into the slope - intercept form.
We get $y=4x - 3$.

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.
Here, $m=-2$, $x_1 = 2$ and $y_1=-1$.

Step2: Substitute values and simplify

Substitute into the point - slope form: $y-(-1)=-2(x - 2)$
Simplify the left - hand side: $y + 1=-2(x - 2)$
Expand the right - hand side: $y+1=-2x + 4$
Subtract 1 from both sides: $y=-2x+3$

Step1: Calculate the slope

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. Let $(x_1,y_1)=(-2,3)$ and $(x_2,y_2)=(1,1)$. Then $m=\frac{1 - 3}{1-(-2)}=\frac{-2}{3}=-\frac{2}{3}$

Step2: Use point - slope form

Using the point $(1,1)$ and $m =-\frac{2}{3}$ in the point - slope form $y - y_1=m(x - x_1)$
We have $y - 1=-\frac{2}{3}(x - 1)$
Expand: $y-1=-\frac{2}{3}x+\frac{2}{3}$
Add 1 to both sides: $y=-\frac{2}{3}x+\frac{2}{3}+1=-\frac{2}{3}x+\frac{5}{3}$

For the line passing through $(0, - 4)$ and $(4,4)$:

Step1: Calculate the slope

Using the slope formula $m=\frac{y_2 - y_1}{x_2 - x_1}$ with $(x_1,y_1)=(0,-4)$ and $(x_2,y_2)=(4,4)$, we get $m=\frac{4-(-4)}{4 - 0}=\frac{8}{4}=2$

Step2: Use slope - intercept form

Since the y - intercept $b=-4$ (because the line passes through $(0,-4)$), the equation is $y = 2x-4$

Answer:

$y = 4x-3$

Problem 1b: