How to Solve Linear Equations in Systems of Equations

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.

Things You'll Need

• Pencil
• Paper
• Calculator

Instructions

Solving A System of Linear Equations

1. 

   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.

2. 

   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.

3. 

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

4. 

   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.

Tips & Warnings

• Write out each step as you go to keep track of your work. It will be easier to find your errors if you have to backtrack.

• When choosing which equation to solve for x or y in the first step, always pick the one that is in the easiest form. If one of the equations already has the x or y by itself, it will probably be the easiest to start with.

• Although most algebra problems use the variables x and y, remember that other letters can be used. Don't always expect to see x and y.

• Unless the numbers are difficult or large, try to work the problem without using a calculator. Then use the calculator to check your answer or solution at the end.