How to Solve: Direct Variation

How to Solve: Direct Variation

Introduction

"If you’ve ever wondered how your phone bill changes when you stream more hours, or how far you travel when you drive faster—direct variation is the math behind it. Master this, and you’ll solve real-world problems AND exam questions in seconds."

What You Need To Know First

Before diving into direct variation, make sure you understand:
1. Proportional relationships – How two quantities increase or decrease together.
2. Solving for one variable – Rearranging equations like y = 3x to find x or y.
3. Basic algebra – Substituting values into equations and simplifying.

Key Vocabulary

Formulas To Know

1. Direct Variation Equation

Formula: y = kx - y = dependent variable (what changes based on x) - x = independent variable (what you control) - k = constant of variation (the fixed rate) MEMORISE THIS – It’s the foundation of every direct variation problem.

2. Finding the Constant (k)

Formula: k = y/x - Used when you have one pair of values to find k. MEMORISE THIS – You’ll use it in almost every problem.

3. Proportion Form (Alternative)

Formula: y₁/x₁ = y₂/x₂ - Used when comparing two sets of values. Given on exam sheet (but you should still know how to use it).

Step-by-Step Method

Follow these steps exactly for every direct variation problem.

Step 1: Identify the Variables

• Read the problem. What two quantities are changing together?
• Label them as y (dependent) and x (independent).

Step 2: Write the Direct Variation Equation

• Start with y = kx.
• If the problem gives you a relationship (e.g., "y varies directly with x"), this is your equation.

Step 3: Find the Constant (k)

• If you’re given a pair of values (x and y), plug them into k = y/x.
• If you’re given a word problem, look for a sentence like "When x = 3, y = 12" to find k.

Step 4: Rewrite the Equation with k

• Once you have k, plug it back into y = kx.
• Now your equation is ready to use.

Step 5: Solve for the Unknown

• If you need to find y for a new x, plug x into y = kx.
• If you need to find x for a new y, rearrange to x = y/k.

Step 6: Check Your Answer

• Does it make sense? If x increases, y should increase too.
• Plug your answer back into the equation to verify.

Worked Example Using the Steps

Problem: y varies directly with x. When x = 4, y = 20. Find y when x = 7.

Step 1: Identify variables. - y depends on x.

Step 2: Write the equation. - y = kx

Step 3: Find k. - Given x = 4, y = 20. - k = y/x = 20/4 = 5

Step 4: Rewrite the equation. - y = 5x

Step 5: Solve for y when x = 7. - y = 5(7) = 35

Step 6: Check. - If x increases from 4 to 7, y should increase from 20 to 35. ✔️

Answer: y = 35

Worked Examples

Example 1 – Basic

Problem: If y varies directly with x, and y = 15 when x = 3, find y when x = 8.

Step 1: y = kx Step 2: k = y/x = 15/3 = 5 Step 3: y = 5x Step 4: y = 5(8) = 40

What we did and why: - We found k using the given values, then used it to find y for a new x. This is the most straightforward direct variation problem.

Example 2 – Medium (Missing x)

Problem: The cost (C) of apples varies directly with the number of pounds (p). If 5 pounds cost $12, how many pounds can you buy for $36?

Step 1: C = kp Step 2: k = C/p = 12/5 = 2.4 Step 3: C = 2.4p Step 4: Rearrange to p = C/k = 36/2.4 = 15

What we did and why: - We found k first, then rearranged the equation to solve for p (pounds) instead of C (cost). This is common when the question asks for the independent variable.

Example 3 – Exam Style (Disguised)

Problem: A car’s fuel efficiency (miles per gallon) is constant. If the car travels 240 miles on 8 gallons, how many gallons are needed for 420 miles?

Step 1: Let m = miles, g = gallons. - m = kg (miles vary directly with gallons) Step 2: k = m/g = 240/8 = 30 (miles per gallon) Step 3: m = 30g Step 4: Rearrange to g = m/k = 420/30 = 14

What we did and why: - The problem didn’t say "direct variation," but the word "constant" told us it was. We treated miles and gallons like y and x, found k, then solved for the unknown.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in. Direct variation means two things change together at a constant rate—like miles and gallons, or hours worked and pay. The magic equation is y = kx. Here’s how to crush it every time:

1. Write the equation: y = kx.
2. Find k: Use k = y/x with the given values.
3. Plug k back in: Now you have y = (your k)x.
4. Solve for what’s missing: Need y? Multiply k by x. Need x? Divide y by k.
5. Check your answer: If x goes up, y should too. If not, you messed up.

Examiners love to hide this in word problems. Look for words like ‘proportional,’ ‘constant rate,’ or ‘varies directly.’ If you see those, you’re dealing with direct variation. Now go practice—you’ve got this!