This grade 8 slope worksheet answer key solves all 22 problems of the proportional relationships and slope worksheet step by step.
Every solution below shows the slope or unit rate step by step, and many include a check with a second point. Compare your method as well as your answer. Final answers are in bold.
Grade 8 Slope Worksheet Answer Key: problems 1–5
Problem 1: Constant from a table
Divide distance by time: \(\frac{110}{2}=55\), \(\frac{220}{4}=55\), \(\frac{330}{6}=55\). The ratio is constant.
Constant of proportionality: \(55\). Equation: \(y=55x\).
Problem 2: Equation from a price
Unit price: \(\frac{10}{4}=2.5\) dollars per cupcake.
Equation: \(y=2.5x\). For 14 cupcakes: \(y=2.5(14)=35\).
The cost is $35.
Problem 3: Slope from a graph
Use the origin and \((4,6)\): rise \(=6\), run \(=4\).
Slope \(=\frac{6}{4}=\frac{3}{2}\). The line passes through the origin, so the equation is \(y=\frac{3}{2}x\). Check with \((2,3)\): \(\frac{3}{2}(2)=3\).
Problem 4: Negative slope
From \((-3,2)\) to \((3,-2)\): rise \(=-2-2=-4\), run \(=3-(-3)=6\).
Slope \(=\frac{-4}{6}=-\frac{2}{3}\). The line passes through the origin, so the equation is \(y=-\frac{2}{3}x\).
Problem 5: Graph y = mx
\(x=0\): \(y=0\). \(x=4\): \(y=\frac{3}{4}(4)=3\). \(x=8\): \(y=\frac{3}{4}(8)=6\).
Points: \((0,0)\), \((4,3)\), \((8,6)\). Plot them and draw a line through them.
Grade 8 Slope Worksheet Answer Key: problems 6–10
Problem 6: Slope and intercept
- Slope \(-2\), \(y\)-intercept 5.
- Slope \(\frac{1}{3}\), \(y\)-intercept \(-4\).
- Slope \(4\), \(y\)-intercept 0, because \(y=4x\) means \(y=4x+0\).
Problem 7: Graph y = mx + b
The \(y\)-intercept is 1, so plot \((0,1)\). The slope is 2, which is \(\frac{2}{1}\): go right 1 and up 2 to reach \((1,3)\), then again to reach \((2,5)\). Draw the line.
Problem 8: Similar triangles
Small triangle: \(\frac{3}{2}\). The large triangle is a copy scaled by \(\frac{6}{2}=3\), so its rise is \(3\times 3=9\).
Large triangle: \(\frac{9}{6}=\frac{3}{2}\). Rise: 9. The triangles are similar, so rise to run is the same ratio, \(\frac{3}{2}\), in both.
Problem 9: Table versus equation
Bike A: unit rate \(12\) mph. Bike B: \(\frac{28}{2}=\frac{42}{3}=\frac{70}{5}=14\) mph.
Since \(14>12\), Bike B is faster, by 2 mph.
Problem 10: Graph versus equation
Printer A: the point \((5,60)\) gives \(\frac{60}{5}=12\) pages per minute.
Printer B: the slope is 15 pages per minute. Since \(15>12\), Printer B is faster by 3 pages per minute.
Grade 8 Slope Worksheet Answer Key: problems 11–15
Problem 11: Compare with words
Noah: \(\frac{18}{4}=4.5\) gallons per minute.
Sofia: \(\frac{8.4}{2}=4.2\), \(\frac{12.6}{3}=4.2\), \(\frac{21}{5}=4.2\) gallons per minute.
Since \(4.5>4.2\), Noah’s hose is faster by 0.3 gallon per minute.
Problem 12: Slope from two points
Slope \(=\frac{11-3}{5-1}=\frac{8}{4}=2\).
Use \((1,3)\): \(3=2(1)+b\), so \(b=1\). Equation: \(y=2x+1\). Check with \((5,11)\): \(2(5)+1=11\).
Problem 13: Slope with negatives
Rise \(=-6-4=-10\). Run \(=3-(-2)=5\).
Slope \(=\frac{-10}{5}=\)\(-2\).
Problem 14: Proportional or not
The rate of change is constant: \(y\) goes up 3 each time \(x\) goes up 1. But when \(x=0\), \(y=2\), not 0.
The line does not pass through the origin, so it is not proportional. The equation is \(y=3x+2\).
Problem 15: Equation from a graph
The line crosses the \(y\)-axis at \((0,4)\), so \(b=4\).
From \((0,4)\) to \((3,-2)\): slope \(=\frac{-2-4}{3-0}=\frac{-6}{3}=-2\).
Equation: \(y=-2x+4\). Check: \(-2(3)+4=-2\).
Grade 8 Slope Worksheet Answer Key: problems 16–20
Problem 16: Missing coordinate
Slope \(=\frac{9}{6}=\frac{3}{2}\), so \(y=\frac{3}{2}x\).
If \(x=10\): \(y=\frac{3}{2}(10)=15\). If \(y=21\): \(x=21: \frac{3}{2}=21\times \frac{2}{3}=14\).
Answers: \(y=15\) and \(x=14\).
Problem 17: Taxi fare
Equation: \(y=2x+3\). The slope 2 is the cost per mile in dollars. The \(y\)-intercept 3 is the flat fee.
For 7 miles: \(y=2(7)+3=17\). The ride costs $17.
Problem 18: Ramp rise
Slope of the first ramp: \(\frac{2}{24}=\frac{1}{12}\).
The ramps are similar triangles, so the second rise is \(\frac{1}{12}\times 30=\frac{30}{12}=2.5\).
The rise is 2.5 feet.
Problem 19: Gym plans
Gym A: \(y=5x+20\). For 10 visits: \(5(10)+20=70\).
Gym B: \(y=9x\). For 10 visits: \(9(10)=90\).
Gym B is proportional because its graph passes through the origin. Gym A is not, because its \(y\)-intercept is 20.
Problem 20: Savings race
Diego: \(\frac{45}{6}=7.5\), so \(y=7.5x\). Aisha: \(\frac{54}{6}=9\), so \(y=9x\).
Difference: \(9-7.5=1.5\). Aisha saves $1.50 more per week.
Grade 8 Slope Worksheet Answer Key: problems 21–22
Problem 21: Savings with a start
Rate: \(\frac{90-50}{6-2}=\frac{40}{4}=10\) dollars per week.
Use \((2,50)\): \(50=10(2)+b\), so \(b=30\). Equation: \(y=10x+30\). Mei started with $30.
It is not proportional, because the line does not pass through the origin.
Problem 22: Draining a tank
Equation: \(y=-8x+120\). The slope \(-8\) means the water falls 8 gallons each minute. The \(y\)-intercept 120 is the starting amount.
Empty when \(y=0\): \(-8x+120=0\), so \(x=15\). The tank is empty after 15 minutes.
How to use this grade 8 slope worksheet answer key
Read this grade 8 slope worksheet answer key after trying each problem yourself, and redo any problem you missed.
See every chapter on the Grade 8 math page.