How to Solve a Direct Variation Equation


Direct variation equations express the relationship between two quantities that either both increase or both decrease at the same time. For example, consider the relationship between hours worked and gross pay. If the number of hours worked increases, pay also increases. Likewise, if the number of hours worked decreases, pay decreases. To understand how to solve direct variation equations, consider the following word problem: Wayne works 6 hours shampooing dogs and earns $90. How much will he earn if he works 8 hours?

  • Write down the direction variation equation y = kx.

  • Substitute the values of each variable into the equation y = kx. This gives 90 = k(6)

    The first part of a direct variation problem will contain two numbers, which in this problem are the number of hours worked -- 6 -- and the amount of money earned -- $90. The second part of the problem gives only the number of hours worked. For this step, use the information in the part of the problem in which both values are given.

  • Divide both sides of the equation by x to solve for k, the proportionality constant.

    For the equation 90 = k(6), divide both sides by 6.

    90/6 = k(6) / 6.

    15 = k.

  • Rewrite the direct variation equation using the value of the proportionality constant, which in this example is 15.

    The equation becomes y = 15x.

  • Substitute the value given in the second part of the problem into the new equation. Wayne worked 8 hours shampooing dogs. Thus, replace x with 8.

    y = 15x becomes y = 15(8).

  • Perform the indicated computations. In this example, you are asked to multiply 15 by 8.

    y = 15(8) = 120.

    Therefore, Wayne earns $120 if he washes dogs for 8 hours.

Tips & Warnings

  • When substituting into the proportionality equation, you may wonder how to know where to put the numbers; whether 6 is equal to x and 90 is equal to y or vice versa. The answer is that it does not matter as long as you are consistent. If you make the number of hours worked equal to x in the first equation, then you must make the number of hours worked equal to x in the second equation. Similarly, if you make the number of hours worked equal to y in the first equation, the number of hours worked must equal y in the second equation.
  • Do not confuse direct variation with inverse variation. In an inverse variation problem, one quantity will increase while the other will decrease. For example, the more someone exercises, the less he will weigh. This is an example of inverse variation. There is a different equation for solving inverse variation problems.


  • Photo Credit Apple Tree House/Lifesize/Getty Images
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