How to Use Linear Combinations to Solve the System of Linear Equations

A linear equation is the equation of a straight line and the degree to which it ascends or descends on a graph. A system of linear equations is just a set of lines that share the same variables. Solving a system of linear equations through linear combination means adding the lines together and solving for their variables. The linear combination method of solving a system of linear inequalities can highlight the relationship of the individual lines.

Instructions

1. Write the equations with one on top of the other. For example, if the equations are 3y + 2x = 5 and x - 5y = -17, then write them as

   3y + 2x = 5

   x - 5y = -17.

2. Rearrange the terms of the equations so that like terms match up. For the example, switch the first two terms in the first equation so that the x-terms and y-terms add up. So

   3y + 2x = 5 becomes 2x + 3y = 5 and the two equations will read

   2x + 3y = 5

   x - 5y = -17.

3. Multiply an entire equation by a number to create a term that mirrors a term in the other equation. For the example, multiplying the second equation by -2 will make the x-term -2x and opposite from the x-term 2x in the other equation, resulting in

   -2x + 10y = 34.

4. Combine the two equations together by addition. For the example, combining

   2x + 3y = 5 and -2x + 10y = 34 results in 13y = 39. The 2x and -2x terms cancel each other out.

5. Solve for the remaining variable. For the example, 13y = 39 and y = 3.

6. Plug the variable into either starting equation. For the example, plugging y = 3 into x - 5y = -17 results in x - 15 =-17 and x = -2.

References

• Math Guide: Systems of Linear Equations
• Math'n Stuff: Use Linear Combination to Solve Systems of Equations and Inequalities