# Solving Systems of Equation by Substitution and Elimination

The problems with the graphing method are threefold:

- You need an accurate graph.

- Your graph may not be large enough.

- It’s hard to estimate the solution if the
coordinates are not integers.

In this section, we look at two algebraic methods for finding
solutions.

- In both of the methods outline below, there are actually
three possible outcomes.

- You get a single ordered pair as a solution.

In this case, the solution is the ordered pair you find.

- All variables go away and you get a false statement, such
as 0 = 4.

In this case, you have parallel lines, so there is no solution
to the system.

- All variables go away and you get a true statement, such
as 0 = 0 or 5 = 5.

In this case, you have the same line, so there are infinitely
many solutions.

## Substitution

**Procedure: (Substitution Method)**

0. Choose a variable and an equation.

1. Solve for the chosen variable in the chosen equation.

2. Substitute the expression you found for the selected
variable in the OTHER equation.

3. Solve the resulting equation in one variable.

4. Use the answer you found in 3 to find the value of the
other variable.

5. Write your answer as an ordered pair.

**Example:**

x + y = 8

2x - 3y = -9

**Answer: **

(3, 5).

## Elimination

**Procedure: (Elimination Method)**

0. Choose a variable.

1. Multiply one or both equations by whatever is necessary to
get the coefficients of the selected variable to be the same, but
with opposite signs.

2. Add the equations together. (NOTE: This eliminates the
selected variable.)

3. Solve the resulting equation in one variable.

4. Use the answer you found in 3 to find the value of the
other variable.

5. Write your answer as an ordered pair.

**Example:**

x + y = 8

3x - y = 0

**Answer:**

(2, 6).