The statement " <span>y </span>varies directly as <span>x </span>," means that when <span>x </span>increases,<span>y </span>increases by the same factor. In other words, <span>y </span>and <span>x </span>always have the same ratio:
<span><span> = k</span> </span>
<span>where </span><span>k </span>is the constant of variation.
<span>We can also express the relationship between </span><span>x </span><span>and </span><span>y </span>as:
<span><span>y = kx</span> </span>
<span>where </span><span>k </span>is the constant of variation.
Since <span>k </span>is constant (the same for every point), we can find <span>k </span>when given any point by dividing the y-coordinate by the x-coordinate. For example, if <span>y </span>varies directly as <span>x </span>, and <span>y = 6</span> when <span>x = 2</span> , the constant of variation is <span>k = = 3</span> . Thus, the equation describing this direct variation is <span>y = 3x </span>.
Example 1: If <span>y </span>varies directly as <span>x </span>, and <span>x = 12</span> when <span>y = 9</span> , what is the equation that describes this direct variation?
<span>k = = </span>
<span>y = x</span>
Example 2: If <span>y </span>varies directly as <span>x </span>, and the constant of variation is <span>k = </span>, what is <span>y </span>when <span>x = 9</span> ?
<span>y = x = (9) = 15</span>
As previously stated, <span>k </span>is constant for every point; i.e., the ratio between the <span>y </span>-coordinate of a point and the <span>x </span>-coordinate of a point is constant. Thus, given any two points <span>(x 1, y 1)</span> and <span>(x 2, y 2)</span> that satisfy the equation, <span> = k </span>and <span> = k </span>. Consequently, <span> = </span>for any two points that satisfy the equation.
Example 3: If <span>y </span>varies directly as <span>x </span>, and <span>y = 15</span> when <span>x = 10</span> , then what is <span>y </span>when <span>x = 6</span> ?
<span> = </span>
<span> = </span>
<span>6() = y </span>
<span>y = 9</span>
