Learning Objectives:

__Math Tutoring on Solving Systems of Three Equations with Three Unknowns__

To solve this **system of equations** there are three different ways:

**1) Substitution, 2) Elimination, 3) Matrices. **

The **process of substitution** involves several steps:

**Step 1:** Solve for one of the variables in one of the equations. It makes no difference which equation and which variable you choose. Let us solve for **x** in equation **(1). **

**Step 2: **Substitute this value for **x** in equations **(2)** and **(3).** This will change equations **(2)** and **(3)** to equations in the two variables **y ****and ****z.** Call the changed equations **(4)** and **(5),** respectively.

**Step 3: **Solve for **y** in equation **(5). **

**Step 4:** Substitute this value of **y **in equation **(4).** This will give you an equation in **z.** Solve for **z. **

**Step 5: **Substitute this value of **z** in equation **(6)** and solve for **y.**

**Step 6:** Substitute for **y** and **z **in equation **(1)** and solve for **x. **

The **process of elimination** involves several steps:

First, reduce three equations to two equations with two variables, and then to one equation with one variable. The **process of using matrices** is essentially a shortcut of the **process of elimination.** Each row of the matrix represents an equation and each column represents coefficients of one of the variables.

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**Hook Questions**:

*1. **What are the three methods to solve the system of three equations with three unknowns?*

*2. **Which method is most convenient and why?*

*3. **Can you solve the system of equations involving more than three unknowns? *

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