Direct Variation

When y varies directly with x, y equals k times x for some fixed number k, called the constant of variation. You can find k from any one pair of values. Here y is twenty four when x is six, so twenty four equals k times six, and k is twenty four divided by six, which is four. The rule is y equals four x. Now use it: when x is fifteen, y is four times fifteen, which is sixty. Every direct variation question works this way. Find k, then use the rule.

To test a table for direct variation, divide each y by its x. In Table A, six over two, fifteen over five, twenty four over eight and thirty three over eleven are all three, so y equals three x. In Table B, the ratios are five, three and a half, three, and two point seven five. They keep changing, so this is not direct variation, even though y goes up by two each time. Table B is y equals two x plus three. A quick test: when x doubles, y must double too. In Table B, x doubles from one to two, but y goes from five to seven.

The graph of y equals k x is a straight line that passes through the origin, and its gradient is k. The line y equals one point five x goes through zero, zero, then through two, three and four, six. Each time x goes up by one, y goes up by one point five. The line y equals x plus two is also straight, but it crosses the y axis at two, not at the origin, so it is not direct variation. When x doubles from two to four, y goes from four to six, which is not double. Both tests must pass: straight, and through the origin.

Sometimes you are given y and asked for x. A car's distance d varies directly with time t, and d is one hundred and fifty when t is two. So k is one hundred and fifty divided by two, which is seventy five, and the rule is d equals seventy five t. When d is three hundred and seventy five, solve three hundred and seventy five equals seventy five t: t is five. There is a quicker way. Three hundred and seventy five is two and a half times one hundred and fifty. In direct variation, if d is two and a half times bigger, t is too, so t is two times two and a half, which is five.

Some quantities vary with the square of another, like the area of a shape and its length. If y varies directly with x squared, then y equals k x squared. With y equal to twelve when x is two, k is twelve divided by two squared, twelve over four, which is three. So y equals three x squared, and when x is five, y is three times twenty five, which is seventy five. Look at the table. When x doubles, y is multiplied by four, which is two squared. When x triples, y is multiplied by nine, which is three squared. Quiz and exercises now.