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Proportional Relationships
as Equations

Learn to write equations like \( y = kx \) that capture proportional relationships between two quantities using the constant of proportionality.

7th Grade Math

What Is a Proportional Relationship?

A proportional relationship exists between two quantities when they increase or decrease at the same constant rate. If you double one quantity, the other doubles too. If you triple one, the other triples.

Constant Rate
The ratio between the two quantities never changes.
Through the Origin
The graph always passes through \( (0, 0) \).

The Constant of Proportionality \( k \)

The constant of proportionality (often called \( k \)) is the unchanging ratio between two proportional quantities. It tells you how much \( y \) changes for every unit change in \( x \).

Items (x) Cost (y)
\( \frac{y}{x} = k \)
1 $3 3 4 $12 3 7 $21 3

Visualizing with a Straight Line

When you graph a proportional relationship, the result is always a straight line through the origin \( (0, 0) \). The steepness of the line equals the constant \( k \).

Items (x) Cost (y) 0 1 2 3 4 3 6 9 12 (0, 0) (1, 3) (2, 6) (3, 9) (4, 12)
\( y = 3x \)

The Core Equation: \( y = kx \)

Every proportional relationship can be written as a single equation. The dependent variable equals the constant of proportionality times the independent variable.

y = k · x dependent constant independent
\( x \)
The independent variable: the input you control.
\( k \)
The constant of proportionality: the unit rate.
\( y \)
The dependent variable: the output that responds.

Finding \( k \) from a Table

To find the constant of proportionality from a table, divide any \( y \)-value by its corresponding \( x \)-value. If the relationship is truly proportional, every row gives the same \( k \).

1
Pick any row from the table (except where \( x = 0 \)).
2
Divide \( y \div x \) to get \( k \).
3
Verify with another row: does \( y \div x \) give the same \( k \)?
4
Write the equation as \( y = kx \).

Proportional vs. Non-Proportional

Not every linear relationship is proportional. The difference comes down to one critical question: does the line pass through the origin?

Proportional Non-Proportional
Equation form \( y = kx \) \( y = mx + b \), \( b \neq 0 \)
Passes through (0, 0)? Always yes No
Ratio \( y/x \) Constant for all rows Changes row to row
Example \( y = 5x \) \( y = 5x + 2 \)

How to Write the Equation

1
Identify the two quantities and decide which is independent (\( x \)) and which is dependent (\( y \)).
2
Check for proportionality: Does the relationship pass through \( (0, 0) \)? Is the ratio \( y/x \) constant?
3
Calculate \( k \): Use \( k = \frac{y}{x} \) from any known pair.
4
Write the equation: Substitute \( k \) into \( y = kx \). Use context-appropriate variable names.
5
Verify: Plug in another pair to confirm the equation works.

Example 1: Apples at the Store

Apples cost $2 per pound. Write an equation for the total cost \( c \) in terms of pounds \( p \).

Pounds (p) Cost (c) 0 1 2 3 4 $2 $4 $6 $8
\( c = 2p \)

Example 2: Distance and Time

A car travels at a constant speed. In 2 hours it covers 110 miles. In 5 hours it covers 275 miles. Write an equation for distance \( d \) in terms of time \( t \).

1
Find \( k \): \( k = \frac{d}{t} = \frac{110}{2} = 55 \)
2
Verify: \( \frac{275}{5} = 55 \) ✓ Same constant!
3
Write equation: \( d = 55t \)

Example 3: From a Graph

The graph below shows the relationship between hours worked and money earned. Write the equation.

Hours (h) Earnings (e) 0 1 2 3 4 5 6 7 $10 $20 $30 $40 $50 (2, 15) (4, 30) (6, 45)

Using Meaningful Variables

While \( y = kx \) is the general form, real-world problems use letters that match the context. This makes the equation easier to read and understand.

\( t = pn \)
t: total cost, p: price per item, n: number of items
\( d = rt \)
d: distance, r: rate (speed), t: time
\( c = 0.15w \)
c: cost, w: weight in pounds, at $0.15/lb

Why Does This Matter?

Proportional equations are the foundation of almost every real-world calculation involving rates, prices, and scaling.

Shopping
Calculating total costs from unit prices at the grocery store.
Cooking
Scaling recipes up or down while keeping ingredient ratios the same.
Science
Relating speed, density, and many physical measurements.

Common Mistakes to Avoid

✕ Forgetting the Origin
Assuming any straight-line relationship is proportional. If \( (0, 0) \) is not on the line, it is not proportional.
✕ Dividing in the Wrong Order
Computing \( k = \frac{x}{y} \) instead of \( k = \frac{y}{x} \). The constant is always \( y \) divided by \( x \).
✕ Not Checking All Rows
Finding \( k \) from one row only. If a second row gives a different \( k \), the relationship is not proportional.

Quick Reference Summary

Concept Key Fact Example
General form \( y = kx \) \( y = 4x \)
Find \( k \) \( k = \frac{y}{x} \) \( k = \frac{20}{5} = 4 \)
Graph Straight line through \( (0,0) \) Slope = \( k \)
Table test All rows give same \( \frac{y}{x} \) \( \frac{8}{2} = \frac{20}{5} = 4 \)
Context variables Use letters that match \( d = 60t \), \( c = 3n \)

Question 1 of 5

A store sells pencils for $0.50 each. Which equation represents the total cost \( c \) for \( n \) pencils?

Question 2 of 5

A table shows: \( (2, 14) \), \( (5, 35) \), \( (8, 56) \). What is the constant of proportionality \( k \)?

Question 3 of 5

Which of the following is not a proportional relationship?

Question 4 of 5

A runner covers 12 miles in 1.5 hours at a constant speed. Which equation gives the distance \( d \) after \( t \) hours?

Question 5 of 5

If \( y = 9x \), what is the value of \( y \) when \( x = 7 \)?