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System of Equations: Addition


A system of equations is a set of equations that are linked together. The simplest system of equations is two equations with two unknowns.

One way to solve a system of equations is by addition:

  1. Prepare the equations so that there is one variable in each equation that has the same coefficient.
    This can be done by multiplying one or both equations by a constant.
  2. Use addition or subtraction to eliminate one of the variables.


Example: Addition

Solve. Answer in the form (x,y). For example: (-2,3)
x - y = 6
-x + 2y = - 9
  Answer (x,y):

Step 1: Prepare the equations
When you add x from the first equation and -x from the second equation, the variable x is eliminated.
So nothing needs to be done to prepare the equations.

Step 2: Add the two equations
x - y = 6
-x + 2y = - 9

y = - 3

Now solve for x.
x - y = 6
x - (-3) = 6
x + 3 = 6
x + 3 - 3 = 6 - 3
x = 3

Answer (x,y):


Example: 2 Equation preparation with subtraction

Solve. Answer in the form (x,y). For example: (-2,3)
x - 2y = 21
4x - 5y = 54
  Answer (x,y):

Step 1: Prepare the equations
Since the coefficients do not match for either variable in the two equations, we will need to use multiplication.
Let's multiply the first equation by 4 to get

4x - 8y = 84
4x - 5y = 54

Step 2: Subtract
4x - 8y = 84
- (4x - 5y = 54)

 
4x - 8y = 84
-4x + 5y = -54

-3y = 30
    y = -10

Now solve for x.
x - 2y = 21
x - 2(-10) = 21
x + 20 = 21
x + 20 - 20 = 21 - 20
x = 1

Answer (x,y):

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