Question

example 1: find the slope of the line that passes through (2, -1) and (-4, 5).
$m = \frac{y_{2}-y_{1}}{x_{2}-x_{1}}$ slope formula
$=\frac{5 - (-1)}{-4 - 2}$ $(x_{1},y_{1})=(2,-1),(x_{2},y_{2})=(-4,5)$
$=\frac{6}{-6}=-1$ simplify
the slope of the line is -1.

example 2: find the slope of the line.
find two points on the line with integer coordinates, such as (1, -2) and (3, -3).
divide the difference in the y - coordinates by the difference in the x - coordinates:
$\frac{-3-(-2)}{3 - 1}=-\frac{1}{2}$
the slope of the line is $-\frac{1}{2}$

exercises
find the slope of the line that passes through each pair of points. express as a fraction in simplest form.

1. (4, 7) and (6, 13)
2. (6, 4) and (3, 4)
3. (5, 1) and (7, -3)
4. (5, -3) and (-4, 3)
5. (5, 10) and (-1, -2)
6. (-1, -4) and (-13, 2)
7. (7, -2) and (3, 3)
8. (-5, 9) and (5, 5)
9. (4, -2) and (-4, -8)

determine the rate of change of each graph.
10.
11.
12.
13.
14.
15.

Explanation:

Step1: Recall slope formula

The slope formula for two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Solve for problem 1

For points $(4,7)$ and $(6,13)$, substitute $x_1 = 4,y_1=7,x_2 = 6,y_2 = 13$ into the formula:
$m=\frac{13 - 7}{6 - 4}=\frac{6}{2}=3$.

Step3: Solve for problem 2

For points $(6,4)$ and $(3,4)$, substitute $x_1 = 6,y_1 = 4,x_2=3,y_2 = 4$ into the formula:
$m=\frac{4 - 4}{3 - 6}=\frac{0}{-3}=0$.

Step4: Solve for problem 3

For points $(5,1)$ and $(7,-3)$, substitute $x_1 = 5,y_1 = 1,x_2 = 7,y_2=-3$ into the formula:
$m=\frac{-3 - 1}{7 - 5}=\frac{-4}{2}=-2$.

Step5: Solve for problem 4

For points $(5,-3)$ and $(-4,3)$, substitute $x_1 = 5,y_1=-3,x_2=-4,y_2 = 3$ into the formula:
$m=\frac{3-(-3)}{-4 - 5}=\frac{6}{-9}=-\frac{2}{3}$.

Step6: Solve for problem 5

For points $(5,10)$ and $(-1,-2)$, substitute $x_1 = 5,y_1 = 10,x_2=-1,y_2=-2$ into the formula:
$m=\frac{-2 - 10}{-1 - 5}=\frac{-12}{-6}=2$.

Step7: Solve for problem 6

For points $(-1,-4)$ and $(-13,2)$, substitute $x_1=-1,y_1=-4,x_2=-13,y_2 = 2$ into the formula:
$m=\frac{2-(-4)}{-13-(-1)}=\frac{6}{-12}=-\frac{1}{2}$.

Step8: Solve for problem 7

For points $(7,-2)$ and $(3,3)$, substitute $x_1 = 7,y_1=-2,x_2 = 3,y_2 = 3$ into the formula:
$m=\frac{3-(-2)}{3 - 7}=\frac{5}{-4}=-\frac{5}{4}$.

Step9: Solve for problem 8

For points $(-5,9)$ and $(5,5)$, substitute $x_1=-5,y_1 = 9,x_2 = 5,y_2 = 5$ into the formula:
$m=\frac{5 - 9}{5-(-5)}=\frac{-4}{10}=-\frac{2}{5}$.

Step10: Solve for problem 9

For points $(4,-2)$ and $(-4,-8)$, substitute $x_1 = 4,y_1=-2,x_2=-4,y_2=-8$ into the formula:
$m=\frac{-8-(-2)}{-4 - 4}=\frac{-6}{-8}=\frac{3}{4}$.

To find the rate of change (slope) from a graph, we pick two points with integer - coordinates on the line and use the slope formula. Since no specific work is shown for the graph - based problems here, we focus on the point - based problems above.

Answer:

1. $3$
2. $0$
3. $-2$
4. $-\frac{2}{3}$
5. $2$
6. $-\frac{1}{2}$
7. $-\frac{5}{4}$
8. $-\frac{2}{5}$
9. $\frac{3}{4}$