Question

1. write the equation of each line in point - slope form. then graph each line.

a) the line with slope 2 through (-3,4)
b) the line through (2,-1) and (5,4)
c) the line through (-3,2) and (1,3)

1. write the equation of each line in slope - intercept form. then graph each line.

2.4b homework geometry

Explanation:

Step1: Recall point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $m$ is the slope and $(x_1,y_1)$ is a point on the line. The slope - intercept form is $y=mx + b$, where $m$ is the slope and $b$ is the y - intercept.

Step2: Solve part a

Given $m = 2$ and the point $(-3,4)$. Using the point - slope form $y - y_1=m(x - x_1)$, we have $y-4 = 2(x + 3)$.
Expand it to get $y-4=2x + 6$.
Then, rewrite in slope - intercept form: $y=2x+10$.
To graph, start at the y - intercept $(0,10)$. The slope is 2, so for every 1 unit increase in $x$, $y$ increases by 2 units.

Step3: Solve part b

First, find the slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Given $(x_1,y_1)=(2,-1)$ and $(x_2,y_2)=(5,4)$.
$m=\frac{4+1}{5 - 2}=\frac{5}{3}$.
Using the point - slope form with the point $(2,-1)$: $y+1=\frac{5}{3}(x - 2)$.
Expand: $y+1=\frac{5}{3}x-\frac{10}{3}$.
Rewrite in slope - intercept form: $y=\frac{5}{3}x-\frac{10}{3}-1=\frac{5}{3}x-\frac{13}{3}$.
To graph, start at the y - intercept $(0,-\frac{13}{3})\approx(0,-4.33)$. The slope is $\frac{5}{3}$, so for every 3 units increase in $x$, $y$ increases by 5 units.

Step4: Solve part c

Find the slope $m=\frac{y_2 - y_1}{x_2 - x_1}$. Given $(x_1,y_1)=(-3,2)$ and $(x_2,y_2)=(1,3)$.
$m=\frac{3 - 2}{1+3}=\frac{1}{4}$.
Using the point - slope form with the point $(-3,2)$: $y - 2=\frac{1}{4}(x + 3)$.
Expand: $y-2=\frac{1}{4}x+\frac{3}{4}$.
Rewrite in slope - intercept form: $y=\frac{1}{4}x+\frac{3}{4}+2=\frac{1}{4}x+\frac{11}{4}$.
To graph, start at the y - intercept $(0,\frac{11}{4})=(0,2.75)$. The slope is $\frac{1}{4}$, so for every 4 units increase in $x$, $y$ increases by 1 unit.

Answer:

a) Point - slope form: $y - 4=2(x + 3)$, Slope - intercept form: $y=2x + 10$
b) Point - slope form: $y + 1=\frac{5}{3}(x - 2)$, Slope - intercept form: $y=\frac{5}{3}x-\frac{13}{3}$
c) Point - slope form: $y - 2=\frac{1}{4}(x + 3)$, Slope - intercept form: $y=\frac{1}{4}x+\frac{11}{4}$