How to Solve Systems of Algebraic Equations Containing Two Variables

Step-by-Step Guide

1. 1

   Step 1: Move the variables to different sides of the equation.

   This "substitution" method starts out by "solving for x" (or any other variable) in one of the equations.
   
   For example, let's say your equations are 4x + 2y = 8 and 5x + 3y =
   9.
   
   Start by looking just at the first equation.
   
   Rearrange it by subtracting 2y from each side, to get: 4x = 8
   - 2y.
   
   This method often uses fractions later on.
   
   You can try the elimination method below instead if you don't like fractions.
2. 2

   Step 2: Divide both sides of the equation to "solve for x."

   Once you have the x term (or whichever variable you are using) on one side of the equation, divide both sides of the equation to get the variable alone.
   
   For example: 4x = 8
   - 2y (4x)/4 = (8/4)
   - (2y/4) x = 2
   - ½y , Make sure you go back to the other equation, not the one you've already used.
   
   In that equation, replace the variable you solved for so only one variable is left.
   
   For example:
   You know that x = 2
   - ½y.
   
   Your second equation, that you haven't yet altered, is 5x + 3y =
   9.
   
   In the second equation, replace x with "2
   - ½y": 5(2
   - ½y) + 3y =
   9. , You know have an equation with only one variable.
   
   Use ordinary algebra techniques to solve for that variable.
   
   If your variables cancel out, skip ahead to the last step.
   
   Otherwise, you'll end up with an answer for one of your variables: 5(2
   - ½y) + 3y = 9 10 – (5/2)y + 3y = 9 10 – (5/2)y + (6/2)y = 9 (If you don't understand this step, learn how to add fractions.
   
   This is often, but not always, necessary for this method.) 10 + ½y = 9 ½y =
   -1 y =
   -2 , Don't make the mistake of leaving the problem half-finished.
   
   You'll need to plug the answer you got back into one of the original equations, so you can solve for the other variable:
   You know that y =
   -2 One of the original equations is 4x + 2y =
   8. (You can use either equation for this step.) Plug in
   -2 instead of y: 4x + 2(-2) =
   8. 4x
   - 4 = 8 4x = 12 x = 3 , When you plug x=3y+2 or a similar answer into the other equation, you're trying to get an equation with only one variable.
   
   Sometimes, you end up with an equation with no variables instead.
   
   Double check your work, and make sure you are plugging the (rearranged) equation one into equation two, not just back into equation one again.
   
   If you're confident you didn't make any mistakes, you have one of the following results:
   If you end up with an equation that has no variables and isn't true (for instance, 3 = 5), the problem has no solution. (If you graphed both of the equations, you'd see they were parallel and never intersect.) If you end up with an equation without variables that is true (such as 3 = 3), the problem has infinite solutions.
   
   The two equations are exactly equal to each other. (If you graphed the two equations, you'd see they were the same line.)
3. 3

   Step 3: Plug this back into the other equation.

4. 4

   Step 4: Solve for the remaining variable.

5. 5

   Step 5: Use the answer to solve for the other variable.

6. 6

   Step 6: Know what to do when both variables cancel out.

Detailed Guide

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