Direct Variation

When two variables are related in such a way that the ratio of their values always remains the same, the two variables are said to be in direct variation.

In simpler terms, that means if A is always twice as much as B, then they directly vary.

If y varies directly as x, the graph of all points that describe this relationship is a line going through the origin (0, 0) whose slope is called the constant of the variation. That's because each of the variables is a constant multiple of the other.

• Expressing Direct Variation an an Equation

The general form of our sample equation y = 6x is written y = kx, where k is the constant of variation. In other words, the value of k does not change.

• Algebraic Interpretation of Direct Variation

For an equation of the form y = kx, multiplying x by some fixed amount also multiplies y by the SAME FIXED AMOUNT. What does this mean? For example, since the perimeter P of a square varies directly as the length of one side of a square, we can say that P = 4s, where the number 4 represents the four sides of a square and s represents the length of one side.

• Geometric Interpretation of Direct Variation

The equation y = kx is a special case of linear equation y = mx + b, where b = 0. (Note: the equation y = mx + b is the slope-intercept form where m is the slope and b is the y-intercept). Anyway, a line through the origin (0,0) always represents a direct variation between y and x. The slope of this line is the constant of variation. In other words, in the equation y = mx + b, m is the constant of variation.

A relationship between two variables in which one is a constant multiple of the other. In particular, when one variable changes the other changes in proportion to the first. If b is directly proportional to a, the equation is of the form b = ka (where k is a constant). An equation in the form y = kx is a direct variation. The quantities represented by x and y are directly proportional, and k is the constant of variation. You can represent any ratio, rate, or conversion factor with a direct variation. Using a direct variation graph is one way to solve proportions. A relationship between two variables wherein their ratio remains constant. An equation or function expressing such a relationship.