Application of Lines

Direct Variation

Two variables vary directly if they are related by an equation of the form y=kxy = kx, where kk is a real number known as the constant of proportionality.
A direct variation equation is a relationship between two variables that can be written in the form:
y=kxy=kx
The constant of proportionality is the coefficient and nonzero constant kk in the direct variation equation. In a direct variation relationship, one variable is proportional to the other. This means that one variable changes in proportion to, or by the same factor as, the other. So, when the value of xx is multiplied by kk, the value of yy is multiplied by kk. The graph of a direct variation relationship y=kxy=kx is a line that passes through the origin, (0,0)(0,0), and the point (1,k)(1,k). The value of kk is the slope of the line, and yy is proportional to xx. The yy-intercept is zero because the line passes through the origin.
The graph of a direct variation relationship passes through the origin and has slope kk. The value of kk affects the steepness of the line.
Step-By-Step Example
Graphing a Direct Variation Equation
Identify the coefficient of proportionality and then graph the direct variation:
y=34xy = \frac{3}{4}x
Step 1
Use the direct variation equation, where kk is the constant of proportionality.
y=kxy=34x\begin{aligned}y&=kx\\y&=\frac{3}{4}x\\ \end{aligned}
The constant of proportionality in the given equation is 34\frac{3}{4}.
Step 2

The graph of a direct variation passes through the point (0,0)(0,0) because bb is zero.

The constant of proportionality is the slope of the line:
RiseRun=34\frac{\text{Rise}}{\text{Run}}=\frac{3}{4}
Solution

Use the slope to determine a second point on the line. Start at the origin and move up 3 units and right 4 units. Then draw a line through the points.

The graph is a line through the origin with a slope of 34\frac{3}{4}. Each time the xx-value increases by 3 units, the yy-value increases by 4 units.