What Is Direct Variations In Algebras?

Direct variation is a mathematical relationship between two variables where the ratio of y to x never changes. It can be expressed in tables, graphs, equations and word problems. 

(Looking for an “Expert mathematician“? Contact us today!)

A real-world example of direct variation would be a recipe that uses 6 cupcakes but needs 1 cup of flour. As you increase the amount of flour, the number of cupcakes increases as well. 

Likewise, a person’s annual salary varies directly with the number of years they have spent in school. As the number of years increases, so does a person’s salary. 

In algebra, this kind of situation is represented by y=kx, where k is a constant value that stays the same throughout the problem. Often, kk is called the constant of proportionality or the constant of variation. 

The constant of proportionality can be used to find a slope between two data points in a direct variation equation and the y-intercept is the salary of someone with no years of schooling. 

Finding the Constant of Proportionality.

You can quickly determine if an equation or table is a direct variation by plugging in 0 for both x and y values. Then, write an equation for y that is equal to the difference between those two numbers, and the equation will be correct. 

Another way to tell if an equation is a direct variation is by plotting the ordered pair of x and y on a graph. Graphs of direct variation always pass through the origin (which is 0). 

A graph can also be used to determine the constant of proportionality by comparing it to a graph that does not represent the relationship. Compare the graphs below and note that in the first graph, the line does not cross through the origin. 

The second graph is a straight line that passes through the origin, so it represents a direct variation. The graph also shows a negative slope and a y-intercept that violates the definition of direct variation. 

This is why it’s important to know whether a table, graph or equation is a direct variation before you begin solving it. It’s easy to mistake a table or graph as being a direct variation when they do not represent it. 

If a table or graph does not represent direct variation, use substitution to figure out the constant of proportionality. Replace the y with k and the x with a nonzero constant that relates the y to x, such as 7. 

When substituting for a nonzero constant in a direct variation equation, remember that k is a constant value that is a constant for every point on the graph of a direct variation. If a graph does not represent direct variation, the constant of proportionality can be found by dividing the y-coordinate of any given point by the x-coordinate. 

In addition to solving direct variation problems using an equation, you can solve them by calculating the constant of proportionality and writing an equation for the y-intercept. The constant of proportionality can also be used to determine the slope between two data points in a direct proportional relation.