Inverse Variation: Step-by-Step Guide
Two quantities have inverse variation when their product stays the same: xy = k, which can also be written y = k/x. The number k is called the constant of variation. When one quantity gets bigger, the other gets smaller, so that multiplying them always gives k.
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Write the equation. 'y varies inversely as x' means y = k/x, or xy = k.\[y = \frac{k}{x}\quad\text{or}\quad xy = k\]
- Find k. Multiply the x and y from the pair you were given. That product is k.\[k = x\,y\]
- Write the equation with k filled in. Put your value of k into y = k/x.\[y = \frac{k}{x}\;(\text{specific } k)\]
- Use the equation to answer the question. Put in the new x to find y, or put in a new y to find x.\[x_{\text{new}}\,y_{\text{new}} = k\]
Worked example
y varies inversely as x, and y = 8 when x = 3. Find y when x = 12.
Step 1: Write y = k/x.
Step 2: Put in x = 3 and y = 8: 8 = k/3. Multiply both sides by 3: k = 24.
Step 3: The equation is y = 24/x.
Step 4: Now find y when x = 12: y = 24/12 = 2.
Step 5: The answer is y = 2.
Check: x times y should always be 24. 3 x 8 = 24 and 12 x 2 = 24. It is the same.
Tips and common mistakes
- In inverse variation, x times y is always the same number, k.
- The graph of y = k/x is a curve, not a straight line. It never touches either axis.
- x can never be 0, because you cannot divide by 0.
- If x doubles, y gets cut in half. If x triples, y becomes one third.
- Do not mix up the two kinds. Direct variation keeps y divided by x the same. Inverse variation keeps y times x the same.
- If a problem says y varies inversely as the square of x, the equation is y = k/x^2.
- If there is more than one variable, like y = k/(xz), use the one set of numbers you are given to find k.