Solving by Substitution

The second method for solving systems of equations is called substitution. This can be a good method when it is easy to isolate one of the variables in one of the equations. Here are the steps to solve a linear system by substitution:

  1. Solve one of the equations for one of the variables.
  2. Substitute that expression for the variable into the other equation.
  3. Solve for the variable.
  4. Substitute the value found for that variable into one of the equations and solve for the other variable.

Example 2

Solve the following system by substitution:

3x + 4y = 6
x + 2y = 4

Step 1: Solve one equation for either x or y. In this system, the second equation is the easiest to solve for x:

x + 2y = 4

x = - y

Answer: x = 4 - 2y

Step 2: Substitute the expression from step 1 into the first equation:
3(-2y + 4) + 4y = 6

Step 3: Solve for y:

y + + 4y = 6

Answer: -6y + 12 + 4y = 6

Step 4: Substitute solution for y into one of original equations and solve to find x. If y = 3,
x + y = 4
x + 2(3) = 4
x + 6 = 4

x =

Answer: x = -2

Now, check your answer, (3, -2) in both equations:
3x + 4y = 6
3(-2) + 4(3) = 6
-6 + 12 = 6
6 = 6

The ordered pair holds true for both equations, so (-2, -3) is the solution to the system of equations.

 

 

 

 

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