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Algebra Help: Direct Variation

When the ratio between two variables always remains the same, the two variables are in direct variation. If A is always two times B, or the cost of apples is $2 per pound, they directly vary. If I buy four pounds of apples, I pay $8, and this relationship does not change. This article will discuss how to express and interpret direct variation.

Direct Variation in an Equation

If one variable, y, is always two times another, x, it is expressed in the following equation: y/x=2 or y=2x. If y/x=1/3, y=1/3x, meaning that y is always three times smaller than x.

Algebraic Interpretation

When we see an equation in the form of y=kx, we know that multiplying x by an amount also increases y by the same amount. For example, if k=2, and x=3, then y=6. If we multiply x*2 so that x=6, we also multiply y*2 and find that y=12.

Geometric Interpretation

If we think of a square with sides of length x, and we know the equation for the perimeter (y) of a square is y=4x, we have an example of direct variation. If we double the length of a side, x, we also double the perimeter.

Graphing Direct Variation

Equations in the form of y=kx are unique cases of the slope-intercept equation y=mx+b where the y-intercept (b) is 0 and m is the constant of variation. Lines of direct variation always pass through the origin (0,0) and have a slope called the constant of the variation. See the graph of our example, y=2x, below. In this case, the constant or slope is 2, because the ratio of these two variables is constantly two. If a line passes through the origin, you know that the two variables directly vary.

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