# Solving Systems of Equations in 3 Unknowns Using Substitution Method

System of linear equations with three equations in three unknowns can be solved by making two substitutions.

Here is the detailed strategy for how to solve systems of equations with three variables.

1. Step 1: First, isolate one unknown () in one equation.

2. Step 2: Then substitute the expression for that unknown into the two other equations (not the equation that you already used in step 1.)

3. Step 3: Now isolate a different unknown in one of the two new equations (the two equations used in step 2).

4. Step 4: Substitute this expression into the last equation.

5. Step 5: Now, you should have one equation with just one unknown. Isolate that unknown to solve for it.

6. Step 6: Once you solve for one unknown, plug it back into any equation to find the other unknowns.

## Examples of how to solve system of equations with 3 variables using substitution method

#### Example1:

Solve the following system of linear equations.

#### Solution:

Before proceeding to solve the systems, it is preferable to number the three equations.

Step 1: Isolate in the equation

Doing this by moving and with change of sign.

Therefor:

Step 2: Plug this expression in for in the equations and .

Simplify these equations. First distribute the coefficient.

Now group and combine like terms together for each equation.

When one term is a fraction, this requires first finding a common denominator.

Step 3: Isolate in the equation .

Doing this by moving with change of sign.

Therefor:

Step 4: Plug this expression for into the equation

Step 5: Solve for in this equation.

Simplify these equation. First distribute the coefficient .

Now group and combine like terms together for the obtained equation.

Step 6: Use this value of to solve for and .

• Plug into the equation .
• Plug and into the equation .

#### Example2:

Solve the following system of linear equations.

#### Solution:

Before proceeding to solve the systems, it is preferable to number the three equations.

Step 1: Isolate in the equation

Doing this by moving and with change of sign.

Therefor:

Step 2: Plug this expression in for in the equations and .

Simplify these equations. First distribute the coefficient.

Now group and combine like terms together for each equation.

Then,

Step 3: Isolate in the equation .

Doing this by moving with change of sign.

Therefor:

Step 4: Plug this expression for into the equation

Step 5: Solve for in this equation.

Simplify these equation. First distribute the coefficient .

Now group and combine like terms together for the obtained equation.

Step 6: Use this value of to solve for and .

• Plug into the equation .
• Plug and into the equation .