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\( y = kx \) &
Constant of Proportionality

Learn to represent proportional relationships with the equation \( y = kx \) and interpret \( k \) as the constant that connects every input to its output.

Pre-Algebra

What Is a Proportional Relationship?

Two quantities are proportional when they increase or decrease at the same rate. If you double one, the other doubles too. If you triple one, the other triples.

✓ Proportional
Buying apples at \( \$2 \) each: 3 apples cost \( \$6 \), 5 apples cost \( \$10 \).
✕ Not Proportional
A phone plan with a \( \$10 \) base fee plus \( \$2 \) per GB: the ratio changes.

The Ratio Test in a Table

Check every row: divide \( y \) by \( x \). If you always get the same number, that number is your constant of proportionality.

\( x \) \( y \) \( \frac{y}{x} \)
133
263
4123
7213

Meet \( k \): the Constant of Proportionality

In the equation \( y = kx \), the letter \( k \) is called the constant of proportionality. It tells you how many units of \( y \) you get for every 1 unit of \( x \).

Unit Rate
\( k \) is the unit rate: the value of \( y \) when \( x = 1 \).
Slope
\( k \) is also the slope of the line through the origin on a graph.
Multiplier
Multiply any \( x \) by \( k \) and you instantly get \( y \).

The Core Equation: \( y = kx \)

\( x \)
\( y \)
 1 2 3 4 2 4 6 8
\( (0, 0) \)
\( (1, 2) \)
\( (2, 4) \)
\( (3, 6) \)
\( (4, 8) \)
\( y = 2x \)

Three Ways to Find \( k \)

1
From a table: pick any row and compute \( k = \frac{y}{x} \).
2
From an equation: \( y = kx \) already shows \( k \) as the coefficient of \( x \).
3
From a graph: read the \( y \)-value when \( x = 1 \), or compute the slope \( k = \frac{\text{rise}}{\text{run}} \).

Proportional vs Non-Proportional

Proportional Non-Proportional
Equation form\( y = kx \)\( y = mx + b \), \( b \neq 0 \)
GraphLine through (0, 0)Line that misses the origin
Ratio \( \frac{y}{x} \)Always the sameChanges row to row
Table clue\( x = 0 \Rightarrow y = 0 \)\( x = 0 \Rightarrow y = b \neq 0 \)

How to Write the Equation

1
Confirm proportionality: Check that the ratio \( \frac{y}{x} \) is the same for every data pair and \( (0, 0) \) fits.
2
Find \( k \): Pick any pair \( (x, y) \) where \( x \neq 0 \) and compute \( k = \frac{y}{x} \).
3
Write the equation: Substitute \( k \) into \( y = kx \).
4
Interpret: State what \( k \) means in context (e.g., "3 miles per hour").

Example 1: Earning Money

Maya earns money babysitting. Her earnings are shown in the table below. Write an equation and find \( k \).

Hours \( (x) \) Earnings \( (y) \)
2\$24
5\$60
8\$96

Example 2: Reading a Graph

Gallons (\(x\))
Miles (\(y\))
 2 4 6 8 50 100 150 200
\( (1, 25) \)

Example 3: Fractional \( k \)

A recipe uses \( \frac{3}{4} \) cup of flour for every 1 batch of cookies. Write the proportional equation for cups of flour \( (y) \) in terms of batches \( (x) \).

1
Identify the rate: \( \frac{3}{4} \) cup per 1 batch, so \( k = \frac{3}{4} \).
2
Write the equation: \( y = \frac{3}{4}x \).
3
Use it: For 6 batches: \( y = \frac{3}{4} \times 6 = \frac{18}{4} = 4.5 \) cups of flour.

Example 4: Finding \( k \) from the Equation

A scientist records that the distance \( d \) (in metres) a snail crawls is modelled by the equation \( d = 0.03t \), where \( t \) is time in seconds.

What is \( k \)?
The coefficient of \( t \) is \( 0.03 \), so \( k = 0.03 \).
What does \( k \) mean?
The snail travels 0.03 metres per second.

How \( k \) Changes the Graph

\( x \)
\( y \)
 1 2 3 4 1 2 3 4
\( k = 0.5 \)
\( k = 1 \)
\( k = 2 \)

Common Mistakes to Avoid

✕ Forgetting (0, 0)
If \( y \neq 0 \) when \( x = 0 \), the relationship is not proportional: it is \( y = mx + b \).
✕ Flipping the Ratio
Computing \( \frac{x}{y} \) instead of \( \frac{y}{x} \). Remember: \( k = \frac{y}{x} \), output over input.
✕ Adding Instead of Multiplying
Writing \( y = k + x \) instead of \( y = k \times x \). Proportional means multiply, not add.

Quick Reference Summary

Concept Key Fact
Equation form\( y = kx \)
Constant of proportionality\( k = \frac{y}{x} \)
\( k \) is also calledUnit rate or slope
Graph shapeStraight line through (0, 0)
Table testAll \( \frac{y}{x} \) ratios are equal
Non-proportional sign\( y \)-intercept \( \neq 0 \) or ratio changes

Question 1 of 5

A car travels 150 miles on 5 gallons of gas. If the relationship is proportional, what is the value of \( k \) in \( y = kx \)?

Question 2 of 5

Which equation represents a proportional relationship?

Question 3 of 5

The table shows a proportional relationship. What is the missing value of \( y \)?

\( x \)\( y \)
318
530
9?

Question 4 of 5

On a graph of \( y = kx \), where can you read \( k \) directly?

Question 5 of 5

A runner covers \( 2.4 \) km in \( 8 \) minutes at a constant speed. What equation models this, with distance \( d \) in km and time \( t \) in minutes?