How to Solve by Substitution

When you are introduced to systems of equations, the first method you usually learn is how to solve by substitution. Later you will also learn how to solve a sytem using graphing and elimination.

The substitution method is easiest to use when one of your equations has a variable that is isolated (solved for), like y = .

For a colorful video explaining the process click here.

How to solve by substitution video

Solving Systems of Equations by Using the Substitution Method


Written Examples of Solving using Substitution

The example below shows how to solve a system of equations using the substitution method. The two equations are shown at the top.

Solving Steps Explanation  
y = 4x     (equation 1)
x + y = 10   (equation 2)
x + 4x = 10 substitute 4x in for y (from equation 1)
5x = 10 simplify the left hand side
x = 2 divide both sides by 5
y = 4x  go back to equation 1 or equation 2
y = 4(2)   substitute your answer, x = 2
y = 8 simplify
Final Answer:  (2, 8) Write your answer as an ordered pair (x, y)

Checking your Final Answer using the Substitution Method

Before we submit our answer, we want to make sure it is correct. There is no better feeling than knowing that you are right!

Simply take both original equations and write them side by side:

  Solving Steps Equation #1 Equation #2
  Substitute (2,8) y = 4x x + y = 10
  so x = 2, y = 8 8 = 4(2) 2 + 8 = 10
  simplify: 8 = 8 10 = 10

Once you simplify the equations, both sides should be equal to the same number.  In this case, 8 = 8 and 10 = 10, so they both check! That's all there is to it - that's how to solve by substitution.

You should now know how to solve by substitution.  However, you do not want to use this method all the time.  In particular, the elimination method is the easiest to use when both of your equations are in the format:

     x + y = #     That is, you have x first, y second, equal to a constant.

Example:  Solve 2x + 3y = 7 and 4x + y = 9 for x and y.

    Notice how both equations have x first, then y, then the constant?

For more great help, return to free algebra help or the GradeA homepage.



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