Slope & Unit Rate — Practice
Slope is a rate of change wearing a geometric coat: how much y moves for every one step in x.
What these questions cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Rate — A ratio of two different units, like miles per hour or dollars per pound.
- Unit rate — A rate with denominator 1 — how much for exactly one.
- Proportional — Two quantities with a constant ratio: yx = k always.
- Slope — How much y changes for each 1 that x increases.
- Rise — The vertical change, y2 - y1.
- Run — The horizontal change, x2 - x1.
Problem 1
Find the slope through (1, 2) and (5, 10).
Worked answer
2
Rise: 10 - 2 = 8
Run: 5 - 1 = 4
m = 84 = 2
Problem 2
Find the slope through (0, 3) and (4, 9).
Worked answer
3/2
Rise: 9 - 3 = 6
Run: 4 - 0 = 4
m = 64 = 32
Problem 3
Find the slope through (2, 8) and (6, 2).
Worked answer
-3/2
Rise: 2 - 8 = -6
Run: 6 - 2 = 4
m = -64 = -32
Problem 4
What is the slope of the line y = -4?
Worked answer
0
Every point has y = -4
Rise is 0
m = 0
Problem 5
A car travels 210 miles in 3 hours. Find the speed in miles per hour.
Worked answer
70
210 ÷ 3 = 70
70 mph
Problem 6
What is the slope of the line x = 7?
Worked answer
undefined
Run is 0
Dividing by zero is not allowed
The slope is undefined
Problem 7
Find the slope through (-3, 5) and (1, -7).
Worked answer
-3
Rise: -7 - 5 = -12
Run: 1 - (-3) = 4
m = -3
Problem 8
A printer prints 1,260 pages in 18 minutes. Find the rate in pages per minute.
Worked answer
70
1260 ÷ 18 = 70
70 pages per minute
Problem 9
A table shows x: 3, 6, 9 and y: 12, 24, 36. Find the unit rate yx.
Worked answer
4
12 ÷ 3 = 4
24 ÷ 6 = 4 ✓
The unit rate is 4
Problem 10
Which relationship is proportional?
- y = 6x
- y = 6x + 2
- y = 6
- y = x2
Worked answer
y = 6x
y = 6x gives y = 0 when x = 0
The others do not, or are not linear
Problem 11
Line A passes through (0,0) and (4, 10). Line B is y = 2x. Which has the larger slope?
Worked answer
A
Line A: m = 104 = 2.5
Line B: m = 2
2.5 > 2, so A
Problem 12
A line passes through (2, 5) and falls 3 units for every 4 units right. Find the slope.
Worked answer
-3/4
Rise = -3, run = 4
m = -34
Problem 13
A line through (3, k) and (7, 14) has slope 2. Find k.
Worked answer
6
14 - k7 - 3 = 2
14 - k = 8
k = 6
Problem 14
Runner A covers 9 km in 45 minutes; runner B covers 14 km in 80 minutes. Who is faster, in km per minute?
Worked answer
A
A: 9 ÷ 45 = 0.2 km/min
B: 14 ÷ 80 = 0.175 km/min
A is faster
Problem 15
A line through (-2, 1) and (c, 13) has slope 4. Find c.
Worked answer
1
13 - 1c + 2 = 4
12 = 4(c + 2)
3 = c + 2, so c = 1
Problem 16
A table shows x: 2, 5, 8 and y: 7, 17.5, 28. Is it proportional? Answer yes or no.
Worked answer
yes
7 ÷ 2 = 3.5
17.5 ÷ 5 = 3.5
28 ÷ 8 = 3.5
Constant, so yes — and y = 3.5x