Question

find slope the slope of a line is the ratio of change in the y-coordinates (rise) to the change in the x-coordinates (run) as you move in the positive direction.

slope of a line $m=\frac{rise}{run}$ or $m=\frac{y_2-y_1}{x_2-x_1}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of any two points on a nonvertical line

example 1: find the slope of the line that passes through $(-3, 5)$ and $(4, -2)$.
let $(-3, 5)=(x_1, y_1)$ and $(4, -2)=(x_2, y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}$ slope formula
$=\frac{-2-5}{4-(-3)}$ $y_2=-2, y_1=5, x_2=4, x_1=-3$
$=\frac{-7}{7}$ simplify.
$=-1$

example 2: find the value of $r$ so that the line through $(10, r)$ and $(3, 4)$ has a slope of $-\frac{2}{7}$.
$m=\frac{y_2-y_1}{x_2-x_1}$ slope formula
$-\frac{2}{7}=\frac{4-r}{3-10}$ $m=-\frac{2}{7}, y_2=4, y_1=r, x_2=3, x_1=10$
$-\frac{2}{7}=\frac{4-r}{-7}$ simplify.
$-2(-7)=7(4-r)$ cross multiply.
$14=28-7r$ distributive property
$-14=-7r$ subtract 28 from each side.
$2=r$ divide each side by $-7$.

exercises
find the slope of the line that passes through each pair of points.

1. $(4, 9), (1, 6)$
2. $(-4, -1), (-2, -5)$
3. $(-4, -1), (-4, -5)$
4. $(2, 1), (8, 9)$
5. $(14, -8), (7, -6)$
6. $(4, -3), (8, -3)$
7. $(1, -2), (6, 2)$
8. $(2, 5), (6, 2)$
9. $(4, 3.5), (-4, 3.5)$

find the value of $r$ so the line that passes through each pair of points has the given slope.

1. $(6, 8), (r, -2), m=1$
2. $(-1, -3), (7, r), m=\frac{3}{4}$
3. $(2, 8), (r, -4) m=-3$
4. $(7, -5), (6, r), m=0$
5. $(r, 4), (7, 1), m=\frac{3}{4}$
6. $(7, 5), (r, 9), m=6$

Explanation:

Step1: Solve Exercise 1

Let $(4,9)=(x_1,y_1)$ and $(1,6)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{6-9}{1-4}=\frac{-3}{-3}=1$

Step2: Solve Exercise 2

Let $(-4,-1)=(x_1,y_1)$ and $(-2,-5)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-5-(-1)}{-2-(-4)}=\frac{-4}{2}=-2$

Step3: Solve Exercise 3

Let $(-4,-1)=(x_1,y_1)$ and $(-4,-5)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-5-(-1)}{-4-(-4)}=\frac{-4}{0}$, undefined (vertical line)

Step4: Solve Exercise 4

Let $(2,1)=(x_1,y_1)$ and $(8,9)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{9-1}{8-2}=\frac{8}{6}=\frac{4}{3}$

Step5: Solve Exercise 5

Let $(14,-8)=(x_1,y_1)$ and $(7,-6)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-6-(-8)}{7-14}=\frac{2}{-7}=-\frac{2}{7}$

Step6: Solve Exercise 6

Let $(4,-3)=(x_1,y_1)$ and $(8,-3)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{-3-(-3)}{8-4}=\frac{0}{4}=0$

Step7: Solve Exercise 7

Let $(1,-2)=(x_1,y_1)$ and $(6,2)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{2-(-2)}{6-1}=\frac{4}{5}$

Step8: Solve Exercise 8

Let $(2,5)=(x_1,y_1)$ and $(6,2)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{2-5}{6-2}=\frac{-3}{4}=-\frac{3}{4}$

Step9: Solve Exercise 9

Let $(4,3.5)=(x_1,y_1)$ and $(-4,3.5)=(x_2,y_2)$.
$m=\frac{y_2-y_1}{x_2-x_1}=\frac{3.5-3.5}{-4-4}=\frac{0}{-8}=0$

Step10: Solve Exercise 10

Use slope formula: $1=\frac{-2-8}{r-6}$
$1=\frac{-10}{r-6}$
$r-6=-10$
$r=-4$

Step11: Solve Exercise 11

Use slope formula: $\frac{3}{4}=\frac{r-(-3)}{7-(-1)}$
$\frac{3}{4}=\frac{r+3}{8}$
$3\times8=4(r+3)$
$24=4r+12$
$4r=12$
$r=3$

Step12: Solve Exercise 12

Use slope formula: $-3=\frac{-4-8}{r-2}$
$-3=\frac{-12}{r-2}$
$-3(r-2)=-12$
$r-2=4$
$r=6$

Step13: Solve Exercise 13

Use slope formula: $0=\frac{r-(-5)}{6-7}$
$0=\frac{r+5}{-1}$
$r+5=0$
$r=-5$

Step14: Solve Exercise 14

Use slope formula: $\frac{3}{4}=\frac{1-4}{7-r}$
$\frac{3}{4}=\frac{-3}{7-r}$
$3(7-r)=-12$
$21-3r=-12$
$-3r=-33$
$r=11$

Step15: Solve Exercise 15

Use slope formula: $6=\frac{9-5}{r-7}$
$6=\frac{4}{r-7}$
$6(r-7)=4$
$6r-42=4$
$6r=46$
$r=\frac{23}{3}$

Answer:

1. $1$
2. $-2$
3. Undefined
4. $\frac{4}{3}$
5. $-\frac{2}{7}$
6. $0$
7. $\frac{4}{5}$
8. $-\frac{3}{4}$
9. $0$
10. $r=-4$
11. $r=3$
12. $r=6$
13. $r=-5$
14. $r=11$
15. $r=\frac{23}{3}$