Problem 44 - College Algebra

Question

Describe in general terms how to solve a system in three variables.

Step-by-Step Solution

Answer

To solve a system in three variables, start by simplifying the equations. Eliminate one variable from two of the three equations to make a two-variable equation. Solve this equation for definite values and substitute these values back into the original equations to obtain the value of the third variable.

Step 1: Understanding the System

A system of three variables typically consists of three equations. Each equation will have three different variables, typically x, y, and z.

Step 1: Simplify the Equations

Start by simplifying each equation as much as possible. This might include expanding brackets, combining like terms, or moving terms from one side of the equation to the other.

Step 2: Eliminate one Variable

The next step is to eliminate one variable from two of the three equations, resulting in two equations in two variables. This is usually achieved by either substitution or elimination method. In substitution, one variable from one equation is expressed as a function of other variables and then substituted into the other equations. In the elimination method, equations are added or subtracted to cancel one variable.

Step 3: Solve the two-Variable Equation

Now that a two-variable equation is obtained, it can be solved using any of the known methods for two-variable equations: substitution or elimination. The solution will provide definite values for two of the variables.

Step 4: Substitute back to obtain the third Variable

Substitute the obtained values into any of the original three equations to find the value of the third variable. Now, we have the value for all three variables.