Introduction to Systems of Linear Equations: Three Variables

Learning Objectives

By the end of this section, you will be able to:

  • Solve systems of three equations in three variables.
  • Identify inconsistent systems of equations containing three variables.
  • Express the solution of a system of dependent equations containing three variables.
A pile of quarters, dimes, nickels, and pennies.

Figure 1. (credit: “Elembis,” Wikimedia Commons)

John received an inheritance of $12,000 that he divided into three parts and invested in three ways: in a money-market fund paying 3% annual interest; in municipal bonds paying 4% annual interest; and in mutual funds paying 7% annual interest. John invested $4,000 more in municipal funds than in municipal bonds. He earned $670 in interest the first year. How much did John invest in each type of fund?

Understanding the correct approach to setting up problems such as this one makes finding a solution a matter of following a pattern. We will solve this and similar problems involving three equations and three variables in this section. Doing so uses similar techniques as those used to solve systems of two equations in two variables. However, finding solutions to systems of three equations requires a bit more organization and a touch of visual gymnastics.