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.
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.
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.
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 \((0,-3)\), so \(b=-3\).
From \((0,-3)\) to \((2,1)\): slope \(=\frac{1-(-3)}{2-0}=\frac{4}{2}=2\).
Equation: \(y=2x-3\). Check: \(2(2)-3=1\).
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.
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.
See every chapter on the Grade 8 math page.