There are two basic ways to solve n equations with n unknowns: substitution and elimination.  Each of them can have variations.  For example, matrix methods are extensions of the elimination method.  In this section, we will explore substitution.

Substitution Method

Two variables and Two unknowns

Let the variables be x and y.  The substitution method has four basic steps:

1. Solve the first equation for y.
2. Substitute the whole right-hand side of the equation (the one you just wrote) in place of "y" in the second equation. You should use parentheses around the substituted expression to keep everything orderly.
3. Solve this equation for x.
4. Use this value of x in any of the equations (the one you wrote in step 1 is easiest) to get the value of y.

Now I will apply these steps to an example, and I will label them so you can understand what I'm doing at each step.

Example: two variables and two unknowns

x=-2y+1
x = y - 5

Step 1: solve eq. 1 for y
  x=-2y+1
  2y=-x+1
  y=(-x+1)/2

Step 2: substitute value of y in eq. 2
  x = y - 5
  x = ((-x+1)/2) - 5

Step 3: solve step 2's eq. for x
  x = -x/2 + 1/2 - 5
  (3/2)x = -9/2
  3x = -9
  x = -3

Step 4: substitute value of x in step 1's eq.
  y=(-x+1)/2
  y=(-(-3)+1)/2
  y=(3+1)/2
  y=4/2
  y=2

Now you should check your answers in the original equations.

x = -2y + 1
-3 = -2(2) + 1
-3 = -4 + 1
-3 = -3
True.

x = y - 5
-3 = 2 - 5
-3 = -3
True.

Can you see how this procedure can be expanded to work with three equations in three variables?  Click here to see how this is done.

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The webmaster and author of the Math Help site is Graeme McRae.
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