How to Solve Linear Equations by Combination

A system of linear equations can be solved as long as you are presented with at least as many distinct equations as you have unknown variables. In such a case, there are two primary methods of solving the system: substitution and combination. Substitution consists of solving one equation in terms of one variable and substituting the solution for that variable into one of the other equations. Combination consists of adding or subtracting entire equations. In any given situation, either approach will yield the same results, usually with the same amount of work.

Instructions

1. Arrange your equations so that all of the variables are on one side of the equals sign.

2. Write the equations so that all common variables are aligned vertically. For example:

   2x + 3y = 5
   x + y = 3

3. Decide on a variable to eliminate, and multiply an entire equation such that the coefficients of that variable are identical between equations. For example, multiply the second equation by 2 to arrive at:

   2x + 3y = 5
   2x + 2y = 6

   This will allow you to eliminate the variable x.

   In a system of more than two equations and variables, you will need to do this more than once before you solve for a variable.

4. Add or subtract the equations such that the variable with similar coefficients cancels out. For example:

   2x + 3y = 5
   - (2x + 2y = 6)

   This calculation results in the combined equation:

   y = -1

5. Solve for the remaining variable. Here, y = -1.

6. Substitute the value obtained for the first variable into either of the original equations and solve for the second variable. For example:

   2x + 3(-1) = 5
   2x - 3 = 5
   2x = 8
   x = 4

Tips & Warnings

• There will be a number of ways through a combination process; it doesn't matter which variable you solve for first. Follow the route that is most intuitive for you.

References

• MIT: Linear Equations
• Richland Community College: Solving Linear Equations