Direct Variation - Grade 8

1. What is Direct Variation?

Definition: Direct variation is a relationship between two variables where one variable is a constant multiple of the other. When one variable changes, the other changes proportionally.

Mathematical Representation:

If $ y $ varies directly as $ x $, we write: $ y \propto x $

The symbol $ \propto $ means "is proportional to" or "varies directly as"

Direct Variation Equation:

$ y = kx $

where $ k $ = constant of variation (a non-zero constant)

Key Properties:

When $ x $ increases, $ y $ increases by the same factor
When $ x $ decreases, $ y $ decreases by the same factor
The ratio $ \frac{y}{x} $ is always constant (equals $ k $)
When $ x = 0 $, then $ y = 0 $
The graph is a straight line passing through the origin

Real-Life Examples:

Distance and time: At constant speed, distance varies directly with time
Cost and quantity: Cost of items varies directly with the number purchased
Earnings and hours: Total earnings vary directly with hours worked
Circumference and radius: $ C = 2\pi r $ (k = 2π)

2. Find the Constant of Variation

Formula:

$ k = \frac{y}{x} $ where $ x \neq 0 $

Steps to Find k:

Identify the values of $ x $ and $ y $ from the problem
Divide $ y $ by $ x $: $ k = \frac{y}{x} $
Simplify if possible
The result is the constant of variation

Examples:

Example 1: If $ y $ varies directly as $ x $, and $ y = 12 $ when $ x = 3 $, find the constant of variation.

Solution:

$ k = \frac{y}{x} = \frac{12}{3} = 4 $

The constant of variation is 4.

Example 2: If $ y $ varies directly as $ x $, and $ y = 20 $ when $ x = 8 $, find $ k $.

$ k = \frac{20}{8} = \frac{5}{2} = 2.5 $

k = 2.5

Example 3: If $ y = 45 $ when $ x = 9 $, find the constant of variation.

$ k = \frac{45}{9} = 5 $

Example 4: A car travels 150 miles in 3 hours at constant speed. Find the constant of variation.

Let distance = $ y $, time = $ x $

$ k = \frac{150}{3} = 50 $ miles per hour

The constant of variation is 50 mph (the speed).

3. Identify Direct Variation

How to Identify Direct Variation:

A relationship is a direct variation if and only if:

The ratio $ \frac{y}{x} $ is constant for all pairs of values
The equation can be written in the form $ y = kx $ (no added constant)
The graph is a straight line through the origin (0, 0)
When $ x = 0 $, then $ y = 0 $

From a Table:

Calculate $ \frac{y}{x} $ for each row. If all ratios are equal, it's direct variation.

Example 1: Is this direct variation?

x	y	y/x
2	10	5
4	20	5
6	30	5

✓ All ratios equal 5 → This IS direct variation with $ k = 5 $

From Equations:

Equation	Direct Variation?	Reason
$ y = 6x $	Yes ✓	Form $ y = kx $, k = 6
$ y = x + 5 $	No ✗	Has +5 (constant added)
$ y = \frac{2}{3}x $	Yes ✓	Form $ y = kx $, k = 2/3
$ y = x^2 $	No ✗	Not linear (has x²)
$ y = -4x $	Yes ✓	Form $ y = kx $, k = -4
$ y = 3x - 2 $	No ✗	Has -2 (constant subtracted)

From a Graph:

Direct Variation: Straight line passing through the origin (0, 0)

NOT Direct Variation: Line doesn't pass through origin OR curved line

4. Write Direct Variation Equations

Steps:

Find the constant of variation $ k = \frac{y}{x} $ using given values
Write the equation $ y = kx $
Substitute the value of $ k $ into the equation

Examples:

Example 1: $ y $ varies directly as $ x $, and $ y = 18 $ when $ x = 6 $. Write the direct variation equation.

Step 1: Find $ k $: $ k = \frac{y}{x} = \frac{18}{6} = 3 $

Step 2: Write equation: $ y = kx $

Step 3: Substitute: $ y = 3x $

Equation: $ y = 3x $

Example 2: If $ y = 24 $ when $ x = 8 $, write the equation showing direct variation.

Find $ k $: $ k = \frac{24}{8} = 3 $

Equation: $ y = 3x $

Example 3: The cost $ C $ varies directly with the number of items $ n $. If 5 items cost $35, write an equation.

$ k = \frac{C}{n} = \frac{35}{5} = 7 $

Equation: $ C = 7n $

Example 4: If $ y = 50 $ when $ x = 10 $, find the equation of direct variation.

$ k = \frac{50}{10} = 5 $

Equation: $ y = 5x $

5. Write and Solve Direct Variation Equations

General Steps:

Find the constant of variation $ k $ using the given information
Write the direct variation equation $ y = kx $
Substitute the known value to solve for the unknown
Check your answer

Alternative Method (Using Proportions):

$ \frac{y_1}{x_1} = \frac{y_2}{x_2} $

Examples:

Example 1: If $ y $ varies directly as $ x $, and $ y = 12 $ when $ x = 4 $, find $ y $ when $ x = 7 $.

Method 1: Using k

Step 1: $ k = \frac{12}{4} = 3 $

Step 2: Equation: $ y = 3x $

Step 3: $ y = 3(7) = 21 $

Answer: y = 21

Method 2: Using proportion

$ \frac{12}{4} = \frac{y}{7} $

$ 4y = 84 $

$ y = 21 $

Example 2: The distance $ d $ a car travels varies directly with time $ t $. If the car travels 120 miles in 2 hours, how far will it travel in 5 hours?

Step 1: $ k = \frac{120}{2} = 60 $ mph

Step 2: $ d = 60t $

Step 3: $ d = 60(5) = 300 $ miles

The car will travel 300 miles in 5 hours.

Example 3: The cost of apples varies directly with their weight. If 3 pounds cost $9, what will 8 pounds cost?

$ k = \frac{9}{3} = 3 $ dollars per pound

Equation: $ C = 3w $

For 8 pounds: $ C = 3(8) = 24 $ dollars

Example 4: If $ y = 30 $ when $ x = 6 $, find $ x $ when $ y = 75 $.

Find $ k $: $ k = \frac{30}{6} = 5 $

Equation: $ y = 5x $

Substitute: $ 75 = 5x $

Solve: $ x = \frac{75}{5} = 15 $

Example 5: The number of pages $ p $ printed varies directly with the time $ t $ in minutes. If 40 pages are printed in 5 minutes, how many minutes will it take to print 100 pages?

$ k = \frac{40}{5} = 8 $ pages per minute

Equation: $ p = 8t $

$ 100 = 8t $

$ t = \frac{100}{8} = 12.5 $ minutes

6. Direct Variation Word Problems

Problem-Solving Strategy:

Read carefully and identify the two variables
Determine if it's direct variation (constant ratio)
Find the constant of variation
Write the equation
Use the equation to answer the question
Include units in your answer

Common Phrases Indicating Direct Variation:

"varies directly as"
"is directly proportional to"
"is proportional to"
"constant rate"
"per unit" (like per hour, per pound, per gallon)

Examples:

Example 1: A recipe for cookies requires 2 cups of flour for every dozen cookies. How many cups of flour are needed for 5 dozen cookies?

This is direct variation: $ F = kD $ (flour varies with dozens)

$ k = \frac{2}{1} = 2 $ cups per dozen

Equation: $ F = 2D $

For 5 dozen: $ F = 2(5) = 10 $ cups

Example 2: A worker earns $15 per hour. How much will they earn in 7 hours?

$ k = 15 $ (dollars per hour)

Equation: $ E = 15h $

$ E = 15(7) = 105 $ dollars

7. Direct Variation vs. Non-Direct Variation

Feature	Direct Variation	NOT Direct Variation
Equation Form	$ y = kx $	$ y = kx + b $ (b ≠ 0)
Graph	Passes through origin	Doesn't pass through origin
When x = 0	y = 0	y ≠ 0
Ratio y/x	Constant	Not constant
Example	$ y = 5x $	$ y = 5x + 3 $