Direct
Proportion
Direct Variation
Directly Proportional
A relationship between two variables in which one is a constant multiple of the other. In particular,
when one variable changes the other changes in proportion
to the first.
If b is directly
proportional to a, the equation is of the form b = ka (where k is
a constant).
Equation: y =
4x
Variable y is directly proportional to x.
Doubling x causes y to double. Tripling x causes y to
triple.
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See
also
Inverse variation, joint
variation, gravity
Worked Example
Problem: The cost of apples varies directly with the number of pounds purchased. If 3 pounds of apples cost $6, how much do 7 pounds cost?
Step 1: Write the direct variation equation, where y is the cost and x is the number of pounds.
Step 2: Substitute the known values (y = 6, x = 3) to find the constant of variation k.
6=k(3)⟹k=36=2 Step 3: Write the specific equation using the value of k.
Step 4: Substitute x = 7 to find the cost of 7 pounds.
y=2(7)=14 Answer: 7 pounds of apples cost $14.
Another Example
This example shows how to verify whether a data set follows direct variation by checking that y/x is constant, rather than solving for an unknown value.
Problem: Determine whether the following table represents a direct variation. If so, find the equation.
| x | y |
|---|---|
| 2 | 10 |
| 5 | 25 |
| 8 | 40 |
Step 1: For a direct variation, the ratio y/x must be the same for every pair of values. Check each ratio.
210=5,525=5,840=5 Step 2: Since all three ratios equal 5, the constant of variation is k = 5.
Step 3: Write the direct variation equation.
Answer: Yes, the table represents a direct variation with the equation y = 5x.
Frequently Asked Questions
What is the difference between direct variation and inverse variation?
In direct variation, y = kx, so as x increases, y increases proportionally. In inverse variation, y = k/x, so as x increases, y decreases. The key test: in direct variation the ratio y/x is constant, while in inverse variation the product xy is constant.
How do you find the constant of variation?
Divide y by x using any known pair of values: k = y/x. If the relationship is truly a direct variation, this ratio will be the same for every pair of (x, y) values in the data set. Once you have k, you can write the full equation y = kx.
Does a direct variation graph always pass through the origin?
Yes. Because y = kx, when x = 0 you get y = 0. The graph of a direct variation is always a straight line through the origin (0, 0). If the line does not pass through the origin, the relationship is not a direct variation even if it is linear.
Direct Variation vs. Inverse Variation
| Direct Variation | Inverse Variation |
|---|
| Equation | y = kx | y = k/x |
| What stays constant | The ratio y/x = k | The product xy = k |
| Graph shape | Straight line through the origin | Hyperbola (two curved branches) |
| Effect of doubling x | y doubles | y is halved |
| Real-world example | Distance traveled at constant speed (d = rt) | Time to finish a job with more workers (t = k/w) |
Why It Matters
Direct variation appears throughout algebra, science, and everyday life. Physics uses it for Hooke's law (spring force varies directly with displacement) and Ohm's law (voltage varies directly with current at constant resistance). Recognizing a direct variation relationship lets you set up a simple equation, make predictions, and solve proportion problems quickly on standardized tests.
Common Mistakes
Mistake: Confusing direct variation with any linear equation. Students sometimes assume y = 3x + 2 is a direct variation.
Correction: A direct variation must have the form y = kx with no added constant. The equation y = 3x + 2 is linear but not a direct variation because its graph does not pass through the origin.
Mistake: Using the wrong ratio when finding k. Some students compute x/y instead of y/x.
Correction: Always isolate k by dividing y by x: k = y/x. Writing k = x/y gives you the reciprocal, which will produce incorrect results when you substitute back into y = kx.
