How to Solve a system of variable equations

Solving a system of equations is taught in algebra.

In algebra, a system of equations is two or more equations with the same set of unknown variables. To solve a system of equations, find the values of all unknown variables that satisfy all of the equations. The elimination method is one of the most commonly used of the multiple ways to solve a system of equations. Elimination requires you to isolate one of the variables by eliminating the other one and solving for the isolated variable.

Instructions

1. Simplify the equations and put them in "Ax +By = C" form if they are not already.

   For example:

   (1/3)x + (1/5)y = 2

   (1/3)x + (1/2)y = -1/2

   Multiply the first equation by the lowest common denominator of 15 and the bottom equation by the lowest common denominator of 6. This simplifies the equations to:

   5x + 3y = 30

   2x + 3y = -3

2. Multiply one or both equations by a number that would allow either x or y to cancel out when the equations are added. In the example, multiplying the second equation by -1 accomplishes this.

   For example:

   5x + 3y = 30

   (-1)(2x + 3y) = (-1)(-3)

   5x + 3y = 30

   -2x - 3y = 3

   The y variables now have opposite coefficients.

3. Add the equations together to get 3x = 33.

   The y variables canceled each other out.

4. Solve for the isolated variable:

   3x = 33

   3x/3 = 33/3

   x = 11

5. Plug in the solved variable to solve for the other variable:

   5x + 3y = 30

   5(11) + 3y = 30

   55 + 3y = 30

   55 + 3y - 55 = 30 - 55

   3y = -25

   y = -25/3

6. Plug the ordered pair (x, y) back into the equations to confirm that it is a valid solution to both of them.

Tips & Warnings

• Systems of equations can contain more than two equations as well as more than two variables.

References

• WTamu.edu: Solving Systems of Linear Equations in Two Variables
• SOSMath: Systems of Equations in Three Variables