Direct Variation

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Direct variation is a relationship where y = kx for some number k that is not 0. The number k is the constant of variation. When x doubles, y doubles too. The graph is a straight line through the origin. For example, in y = 3x, the constant is 3.

Before this lesson: You should know proportions, how to find slope, and how to read slope-intercept form.

What Is Direct Variation?

Say you earn $15 an hour. Work 0 hours and you earn $0. Work 2 hours and you earn $30. Work 4 hours and you earn $60. Your pay is always 15 times your hours.

That is direct variation. One amount is always the same multiple of the other. We say “y varies directly with x.” Another name is “y is proportional to x.”

Because y = kx, dividing gives \tfrac{y}{x} = k. So in direct variation, the ratio \tfrac{y}{x} is the same for every pair of values.

Two lines on a grid from −5 to 5. The blue line y = 2x passes through the origin (0, 0), marked with a dot, so it is direct variation. The orange dashed line y = 2x + 3 has the same slope but crosses the y-axis at (0, 3), so it is not direct variation. A table beside the graph lists x = 1, 2, 3 with y = 2, 4, 6 for y = 2x, where y ÷ x is always 2.
Both lines have slope 2, but only y = 2x passes through the origin. That is why only y = 2x is direct variation.

The Direct Variation Equation

y = kx \qquad k = \frac{y}{x}
  • k is the constant of variation. It is also the slope of the line, and the unit rate.
  • There is no added number. In y = mx + b form, b = 0.
  • The graph always passes through (0, 0).

How to tell if a relationship is direct variation

You are givenIt is direct variation ifExample
An equationit can be written as y = kx3y = 12x becomes y = 4x
A table\tfrac{y}{x} is the same for every pair\tfrac{14}{4} = \tfrac{21}{6} = \tfrac{7}{2}
A graphit is a straight line through the originy = 2x in the graph above

Steps to solve a direct variation problem

  1. Find k by dividing: k = \tfrac{y}{x}.
  2. Write the equation y = kx with your value of k.
  3. Put in the new x or y and solve.

Worked Examples

Example 1: Which equations are direct variation?

  • y = 5x: yes. It is already y = kx with k = 5.
  • y = 5x - 2: no. The -2 means the line misses the origin.
  • 3y = 12x: divide both sides by 3 to get y = 4x. Yes, with k = 4.

Example 2: Is the table direct variation?

x24610
y7142135

Find \tfrac{y}{x} for each pair: \tfrac{7}{2}, \tfrac{14}{4} = \tfrac{7}{2}, \tfrac{21}{6} = \tfrac{7}{2} and \tfrac{35}{10} = \tfrac{7}{2}. All the ratios match.

Answer: Yes. The equation is y = \tfrac{7}{2}x.

Compare a table with x = 1, 2, 3 and y = 4, 7, 10. The ratios are 4, \tfrac{7}{2} and \tfrac{10}{3}. They don’t match, so that table is not direct variation.

Example 3: Find k, then find y

y varies directly with x, and y = 18 when x = 4. Find y when x = 10.

\begin{aligned} k &= \frac{18}{4} = \frac{9}{2} \\ y &= \tfrac{9}{2}x \\ y &= \tfrac{9}{2}(10) = 45 \end{aligned}

Answer: y = 45. Check with a ratio: \tfrac{45}{10} = \tfrac{9}{2}. It matches.

Example 4: A negative constant

y varies directly with x, and y = -12 when x = 8. Find x when y = 9.

\begin{aligned} k &= \frac{-12}{8} = -\frac{3}{2} \\ 9 &= -\tfrac{3}{2}x \\ x &= 9 \cdot \left(-\tfrac{2}{3}\right) = -6 \end{aligned}

Answer: x = -6. With a negative k, one value is positive when the other is negative.

Example 5: Word problem with hourly pay

Ava’s pay varies directly with the hours she works. She earns $78 for 6 hours. Write an equation for her pay P after h hours. How much does she earn for 15 hours?

\begin{aligned} k &= \frac{78}{6} = 13 \\ P &= 13h \\ P &= 13(15) = 195 \end{aligned}

Answer: P = 13h. She earns $195 for 15 hours. The constant, 13, is her pay rate: $13 per hour.

Common Mistakes

MistakeExample of the errorFix
Calling any line direct variationSaying y = 5x - 2 is direct variationThere must be no added number. The line must pass through (0, 0).
Dividing in the wrong ordery = 18 when x = 4: writing k = \tfrac{4}{18}Divide y by x: k = \tfrac{18}{4} = \tfrac{9}{2}.
Checking only one pair in a tableSeeing \tfrac{4}{1} = 4 and stoppingCheck every pair. All the ratios must match.
Adding k instead of multiplyingWriting y = x + 13 for $13 per hourDirect variation multiplies: y = 13x.

Direct Variation and Proportions

Since \tfrac{y}{x} is always the same, you can also solve with a proportion. In Example 3, set the two ratios equal.

\frac{18}{4} = \frac{y}{10} \quad\Rightarrow\quad 4y = 180 \quad\Rightarrow\quad y = 45

This is the same answer as before. Use whichever way you like. The y = kx way is faster when you need many values.

The constant k is also the slope of the line. A larger k makes a steeper line. A negative k makes a line that falls from left to right.

Practice Problems

  1. Is y = -4x direct variation? If so, give k.
  2. Is y = 4x + 1 direct variation?
  3. Is 2y = 9x direct variation? If so, give k.
  4. Is this table direct variation? x: 3, 6, 9 and y: 2, 4, 6. If so, write the equation.
  5. Is this table direct variation? x: 1, 2, 4 and y: 3, 5, 9.
  6. y varies directly with x, and y = 20 when x = 5. Find y when x = 12.
  7. y varies directly with x, and y = 6 when x = 9. Find y when x = 15.
  8. y varies directly with x, and y = -10 when x = 4. Find x when y = 25.
  9. A line passes through (0, 0) and (3, -6). Is it direct variation? If so, write the equation.
  10. Weight on the Moon varies directly with weight on Earth. A 150-pound person weighs 25 pounds on the Moon. How much does an 84-pound dog weigh on the Moon?
Show answers
  1. Yes. k = -4.
  2. No. The +1 means the line does not pass through the origin.
  3. Yes. Divide by 2: y = \tfrac{9}{2}x, so k = \tfrac{9}{2}.
  4. Yes. Every ratio is \tfrac{2}{3}: \tfrac{2}{3}, \tfrac{4}{6}, \tfrac{6}{9}. The equation is y = \tfrac{2}{3}x.
  5. No. The ratios are 3, \tfrac{5}{2} and \tfrac{9}{4}, which don’t match.
  6. k = \tfrac{20}{5} = 4, so y = 4x. Then y = 4(12) = 48.
  7. k = \tfrac{6}{9} = \tfrac{2}{3}, so y = \tfrac{2}{3}x. Then y = \tfrac{2}{3}(15) = 10.
  8. k = \tfrac{-10}{4} = -\tfrac{5}{2}. Then 25 = -\tfrac{5}{2}x, so x = 25 \cdot \left(-\tfrac{2}{5}\right) = -10.
  9. Yes. The line goes through the origin, and k = \tfrac{-6}{3} = -2. The equation is y = -2x.
  10. k = \tfrac{25}{150} = \tfrac{1}{6}, so M = \tfrac{1}{6}E. The dog weighs \tfrac{1}{6}(84) = 14 pounds on the Moon.

Direct Variation FAQ

What is direct variation in math?

It is a relationship written as y = kx, where k is a nonzero constant. The ratio \tfrac{y}{x} stays the same, and the graph is a line through the origin.

How do you find the constant of variation?

Divide y by x for any known pair: k = \tfrac{y}{x}. If y = 18 when x = 4, then k = \tfrac{9}{2}.

Is every linear equation direct variation?

No. Only lines that pass through the origin are. y = 2x is direct variation, but y = 2x + 3 is not.

What is the difference between direct and inverse variation?

In direct variation, y = kx, so y grows when x grows. In inverse variation, y = \tfrac{k}{x}, so y shrinks when x grows. Inverse variation is taught in Algebra 2.

Summary

  • Direct variation means y = kx, where k is a nonzero constant.
  • The constant of variation is k = \tfrac{y}{x}. It is also the slope.
  • The graph is a straight line through the origin.
  • In a table, every pair must have the same ratio \tfrac{y}{x}.
  • To solve a problem, find k, write y = kx, then put in the new value.

Previous lesson: Parallel and Perpendicular Lines

Next lesson: Average Rate of Change (coming soon)

Back to the Algebra 1 Index

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