Direct Variation: Step-by-Step Guide
Two quantities have direct variation when one is always the same number times the other: y = kx. The number k is called the constant of variation. When x gets bigger, y gets bigger by the same factor. The graph is a straight line that goes through the point (0, 0).
Math Racer skill guide · Algebra 1 · Grade 9
How to solve it, step by step
- Write the equation. 'y varies directly as x' means y = kx. We do not know k yet.\[y = kx\]
- Find k. Use the pair of numbers you were given. Put x and y into y = kx and solve for k.\[k = \frac{y}{x}\]
- Write the equation with k filled in. Put your value of k back into y = kx.\[y = k\,x \;(\text{specific } k)\]
- Use the equation to answer the question. Put in the new x to find y, or put in a new y to find x.\[y_{\text{new}} = k\,x_{\text{new}}\]
Worked example
y varies directly as x, and y = 18 when x = 6. Find y when x = 11.
Step 1: Write y = kx.
Step 2: Put in x = 6 and y = 18: 18 = k(6). Divide by 6: k = 3.
Step 3: The equation is y = 3x.
Step 4: Now find y when x = 11: y = 3(11) = 33.
Step 5: The answer is y = 33.
Check: y divided by x should always equal k. 18/6 = 3 and 33/11 = 3. It is the same.
Tips and common mistakes
- In direct variation, y divided by x is always the same number, k.
- The graph of y = kx always goes through (0, 0). If the line does not go through the origin, it is not direct variation.
- k can be positive or negative. If k is negative, y gets smaller as x gets bigger.
- Use the pair of numbers you are given to find k first. Then use k to answer the rest.
- 'y is proportional to x' means the same thing as 'y varies directly as x'.
- If a problem says y varies directly as the square of x, the equation is y = kx^2.
- In word problems, include the units in your answer.