Question

Graph the equation. State whether the two quantities have direct variation. If they have direct variation, find the constant of variation and the slope of the direct variation model. $$y=2 x$$

Short answer

The equation \(y = 2x\) exhibits direct variation. The constant of variation and the slope of the direct variation model is 2.

Step-by-step solution

Understand Direct Variation

Direct variation is when one variable increases or decreases in direct proportion with another. It follows the model y = kx where k is the proportionality constant.

Graph the Equation

Plot the straight line equation y = 2x on a graph. Choose random x-values and substitute them in the equation to get the corresponding y-values. For example, if x = 1, y = 2(1) = 2. So, the point (1,2) is on the graph. Repeat this step with more values for x to get more points on the graph.

Determine Direct Variation

Since the equation follows the form y = kx, where k is a constant, it represents direct variation. Therefore, the equation y = 2x exhibits direct variation.

Find the Constant of Variation

In the equation y = 2x, there is no added or subtracted number on the right side, only the number 2 multiplied by x. Thus, the constant of variation is 2 in both cases..

Find the Slope of the Direct Variation Model

The slope of a direct variation equation is the constant of variation. Therefore, the slope of the equation y = 2x is 2.