How to Do a System of Linear Equations

A system of linear equations is a group of equations that are constructed with the same variables. The variable's values are the same across the equations, and can be divined by manipulations on the individual equations. You can solve a system of linear equations either by substituting equations into each other or by using the basic properties of the equations to eliminate opposite terms. Learning how to work a system of linear equations correctly can help you understand their shared properties.

Instructions

Substitution

1. Solve one of the equations for a variable. For example, if the equations are

   y - x = 1 and 2x - y = 0, then solving the first equation for y would result in y = x + 1.

2. Substitute the value obtained in the first equation into the second equation. For the example, substituting y = x + 1 into 2x - y = 0 results in 2x -( x+1)= 0.

3. Solve the second equation. For the example, 2x -( x+1)= 0 equals 2x - x -1= 0.

   2x - x -1= 0 equals 2x - x= 1, and results in x=1.

4. Plug the value of the variable into either original equation to get the value of the other variable. For the example, plugging the value of x as 1 into the original equation y - x = 1 results in

   y - 1 = 1 and y = 2. The solution of the set is x=1 and y=2.

Addition/Elimination

1. Write the equations stacked upon each other. For example, if the equations are

   y - 2x =4 and y + 2x =8 then write the equations as follows

   y - 2x =4

   y + 2x =8.

2. Add the terms of the equations together like an ordinary addition problem. For the example,

   y - 2x =4

   y + 2x =8

   ------------

   2y=12.

   The 2x and -2x cancel each other out.

3. Solve the equation for the remaining variable. For the example, 2y =12 and y =6.

4. Plug the value of the solved variable into either original equation to obtain the value of the other variable. For the example, plugging the value y=6 into the equation

   y - 2x =4 results in 6 - 2x =4. Then, -2x=-2 and x=1. The solution of the set is x=1 and y=6.

References

• Syracuse University: Office of International Research and Assessment, Solving Systems of Equations by Substitution
• Purple Math: Systems of Linear Equations, Solving by Addition/Elimination