How to Solve Linear Systems by Adding or Subtracting

A system of linear equations is comprised of two linear equations. Each linear equation in the system contains two variables and will form a straight line when drawn onto a coordinate plane. To solve a system, you must find the number that each variable stands for. Variables are the letters in each equation. Those numbers must work in both equations to make them true statements. Variables are represented by letters, such as "x" and "y." Do not get too accustomed to seeing "x" and "y" in algebra problems; some textbooks and teachers like to change up the letters. One way to determine the solution for a system of equations is the addition/subtraction method. The information below will guide you through the steps of this method by using an example system: -10y + 2x = -20; x - 4y = 10.

Things You'll Need

• Pencil
• Paper
• Calculator

Instructions

1. Write the two equations one right underneath the other, just like a regular addition or subtraction problem. Be sure to line up the variables in each equation.-10y + 2x = -20x – 4y = 10-10y + 2x = -20-4y + x = 10

2. Eliminate one variable, either "x" or "y," from the two equations. In this problem, the "x" variable is easiest to eliminate because of the small coefficients. To cancel out the 2x, the "x" in the second equation must be multiplied by -2. Remember that whatever is done to one term in an equation must be done to all terms, so the -4y and 10 must also be multiplied by -2.-10y + 2x = -20-4y(-2) + x(-2) = 10 (-2)-10y + 2x = -20 8y – 2x = -20

3. With the equations still lined up one under another, add/subtract the bottom terms from the like terms directly above them, like a normal addition/subtraction problem. -10y + 2x = -20 8y -2x = -20_____________-2y = -40

4. Solve for the "y" variable. -2y = -40-2y/-2 = -40/-2 y = 20

5. Substitute the newly found "y" value into either of the two original equations in order to find "x." -10y + 2x = -20-10(20) + 2x = -20-200 + 2x = -202x = 180x = 90

6. Check your solutions by inputting the "x" and "y" values into each equation. If working with smaller numbers, try to solve the problem on your own. Then you can use a calculator to double-check your math and to also verify your answers. For this problem, our solution is { (90, 20) } and both values work in each equation.-10(20) + 2(90) = -20-200 + 180 = -20-20 = -20x - 4y = 1090 – 4(20) = 1090-80 = 1010 = 10

Tips & Warnings

• You may also see the addition/subtraction method for solving linear systems as being called the elimination or elimination/addition method.

• There are other ways of solving linear systems, including the graphing method and the substitution method.