# Systems of Equations and Inequalities

## Systems of Linear Equations: Three Variables


### Learning Objectives

1. Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1)

2. Solve a system of three linear equations with three variables (IA 4.4.2)

### Objective 1: Determine whether an ordered triple is a solution of a system of three linear equations with three variables (IA 4.4.1) 

A linear equation with three variables where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form  . The graph of a linear equation with three variables is a plane.

A system of linear equations with three variables is a set of linear equations with three variables. For example,

is a system of linear equations with three variables.
Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an ordered triple (x,y,z).


### Practice Makes Perfect


Determine whether the ordered pairs are solutions to the given system.



### Objective 2: Solve a system of three linear equations with three variables (IA 4.4.2)

When we solve a system of linear equations with three variables, we have many possible solutions.

The solutions are summarized in the table below.


### Practice Makes Perfect

Jordi 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. Jordi invested $4,000 more in mutual funds than in municipal bonds. He earned $670 in interest the first year. How much did Jordi 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 visualization.

### Solving Systems of Three Equations in Three Variables

In order to solve systems of equations in three variables, known as three-by-three systems, the primary tool we will be using is called Gaussian elimination, named after the prolific German mathematician Karl Friedrich Gauss. While there is no definitive order in which operations are to be performed, there are specific guidelines as to what type of moves can be made. We may number the equations to keep track of the steps we apply. The goal is to eliminate one variable at a time to achieve upper triangular form, the ideal form for a three-by-three system because it allows for straightforward back-substitution to find a solution 
  which we call an ordered triple. A system in upper triangular form looks like the following:
The third equation can be solved for 
  and then we back-substitute to find 
  
 and 
  
 To write the system in upper triangular form, we can perform the following operations:
1. Interchange the order of any two equations.
2. Multiply both sides of an equation by a nonzero constant.
3. Add a nonzero multiple of one equation to another equation.
The solution set to a three-by-three system is an ordered triple 
  
 Graphically, the ordered triple defines the point that is the intersection of three planes in space. You can visualize such an intersection by imagining any corner in a rectangular room.  A corner is defined by three planes: two adjoining walls and the floor (or ceiling). Any point where two walls and the floor meet represents the intersection of three planes.

### Identifying Inconsistent Systems of Equations Containing Three Variables

Just as with systems of equations in two variables, we may come across an inconsistent system of equations in three variables, which means that it does not have a solution that satisfies all three equations. The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location. The process of elimination will result in a false statement, such as 
  
 or some other contradiction.

### Expressing the Solution of a System of Dependent Equations Containing Three Variables

We know from working with systems of equations in two variables that a dependent system of equations has an infinite number of solutions. The same is true for dependent systems of equations in three variables. An infinite number of solutions can result from several situations. The three planes could be the same, so that a solution to one equation will be the solution to the other two equations. All three equations could be different but they intersect on a line, which has infinite solutions. Or two of the equations could be the same and intersect the third on a line.

### Key Concepts

1. A solution set is an ordered triple 
  
 that represents the intersection of three planes in space. See .
2. A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated.   The steps include interchanging the order of equations, multiplying both sides of an equation by a nonzero constant, and adding a nonzero multiple of one equation to another equation. See .
3. Systems of three equations in three variables are useful for solving many different types of real-world problems. See .
4. A system of equations in three variables is inconsistent if no solution exists. After performing elimination operations, the result is a contradiction. See .
5. Systems of equations in three variables that are inconsistent could result from three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
6. A system of equations in three variables is dependent if it has an infinite number of solutions. After performing elimination operations, the result is an identity. See .
7. Systems of equations in three variables that are dependent could result from three identical planes, three planes intersecting at a line, or two identical planes that intersect the third on a line.

### Section Exercises


### Verbal


### Algebraic

For the following exercises, determine whether the ordered triple given is the solution to the system of equations.
For the following exercises, solve each system by elimination.
For the following exercises, solve each system by Gaussian elimination.

### Extensions

For the following exercises, solve the system for 
  and 
 


### Real-World Applications
