How to Find Zeros of Linear Functions

By Bob Barber; Updated April 24, 2017
Know the zeros to a linear function when your teacher asks you to identify one.

The zero of a linear function in algebra is the value of the independent variable (x) when the value of the dependent variable (y) is zero. Linear functions that are horizontal do not have a zero because they never cross the x-axis. Algebraically, these functions have the form y = c, where c is a constant. All other linear functions have one zero.

Determine which variable in your function is the dependent variable. If your variables are x and y, y is the dependent variable. If your variables are letters other than x and y, the dependent variable will be the variable that is plotted on a vertical axis (like y).

Substitute zero for the dependent variable in the equation of your function. Don't worry about the form of the equation (standard, slope-intercept, point-slope); it doesn't matter. After substitution, the value of the term, including the dependent variable, becomes zero and drops out of the equation. For example, if your equation is 3x + 11y = 6, you would substitute zero for y, the term 11y would drop out of the equation and the equation would become 3x = 6.

Solve the equation of your function for the remaining (independent) variable. The solution is the zero of the function, which means that it tells where the graph of the function crosses the x-axis. For example, if your equation is 3x = 6 after substitution, you would divide both sides of the equation by 3 and your equation would become x = 2. Two is the zero of the equation, and the point (2, 0) would be where your function crosses the x-axis.

Tip

Another way to think of the dependent variable is that the dependent variable measures the outcome of a real-life situation. For example, suppose you are given a linear function where "f" stands for the amount of food given to fish per week, and "w" stands for the weight of the fish after one month. Even if you are not told so, you would understand in a common-sense way that the investigator would have manipulated the amount of food given to the fish; however, she could not have manipulated the resulting weight of the fish; she could only have measured it. Therefore, "w" would be the dependent (or unmanipulated, or outcome) variable.

Linear equations of the form x = c, where "c" is a constant, are not functions. They are often included in the study of linear functions, however. Graphically, these equations are plotted as vertical lines that cross the x-axis at c. For example, the equation x = 3.5 is a vertical line that crosses the x-axis at the point (3.5, 0).

About the Author

Bob Barber is a freelance writer in Massachusetts. He has a Bachelor of Arts in biology from Yale University and a Master of Science in marine biology from the University of Delaware.