How to Solve Linear Equations in Systems of Equations

By Rachel Morgan

Linear equations are equations that are represented in the form ax + b = 0, in which x is the variable and a and b are real numbers. Most linear equations you will come across in your algebra courses will not be in that perfect form. Each term will be in a different order and the equation may contain a second variable, usually y. Regardless, linear equations will form a straight line when graphed. A system of equations is made up of two linear equations. If you graphed both lines in a system, the point of intersection would represent the solution. Since it is beneficial to see a math problem being worked, the steps to solve a system of linear equations are shown below using the system 9x + 4y = 36 and y + 3x = 9.

Solving A System of Linear Equations

Choose one of the equations in the system to solve for the variable x or y. In the case of the sample problem, let's solve for y.

Substitute the newly found y value for the y variable in the first equation. Use the distributive property to get rid of the parenthesis in the equation and combine like terms in order to solve for x.

Take the newly found x value (0) and substitute it in the second equation of the system to solve for y.

Check the answers for the x and y variables by testing them in both equations. If the values for x and y make each equation true, the solution set for the system has been found. In this sample problem, the solution { (0, 9) } works for both equations in the system.

About the Author

Rachel Morgan began her writing career in 2008 after previously working in her state's community college system. She focuses on health and fitness writing, in addition to blogging for small businesses. An alumna of the University of North Carolina, Morgan has a bachelor's degree in public health and has studied PR in the past.