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Solving Systems of Linear Equations in Three Variables

In this section we use elimination of variables to solve systems of equations in three variables.

Definition

The equation 5x - 4y = 7 is called a linear equation in two variables because its graph is a straight line. The equation 2x + 3y - 4z = 12 is similar in form, and so it is a linear equation in three variables. An equation in three variables is graphed in a three-dimensional coordinate system. The graph of a linear equation in three variables is a plane, not a line. We will not graph equations in three variables in this text, but we can solve systems without graphing. In general, we make the following definition.

Linear Equation in Three Variables

If A, B, C, and D are real numbers, with A, B, and C not all zero, then Ax + By + Cz = D is called a linear equation in three variables.

A solution to an equation in three variables is an ordered triple such as (-2, 1, 5), where the first coordinate is the value of x, the second coordinate is the value of y, and the third coordinate is the value of z. There are infinitely many solutions to a linear equation in three variables.

The solution to a system of equations in three variables is the set of all ordered triples that satisfy all of the equations of the system.

The strategy that we follow for solving a system of three linear equations in three variables is stated as follows.

Solving a System in Three Variables

1. Use substitution or addition to eliminate any one of the variables from a pair of equations of the system. Look for the easiest variable to eliminate.

2. Eliminate the same variable from another pair of equations of the system.

3. Solve the resulting system of two equations in two unknowns.

4. After you have found the values of two of the variables, substitute into one of the original equations to find the value of the third variable.

5. Check the three values in all of the original equations.

Example

Using addition and substitution

Solve the system:

(1) x + y   = 4
(2) 2x   -3z = 14
(3)   2y +z = 2

Solution

From Eq. (1) we get y = 4 - x. If we substitute y = 4 - x into Eq. (3), then Eqs. (2) and (3) will be equations involving x and z only.

(3) 2y + z = 2  
  2(4 - x) + z = 2 Replace y by 4 - x.
  8 - 2x + z = 2 Simplify.
(4) -2x + z = -6  

Now solve the system consisting of Eqs. (2) and (4) by addition:

2x - 3z

-2x + z

= 14

= -6

Eq. (2)

Eq. (4)

-2z

z

= 8

= -4

 

Use Eq. (1) to find x:

x + y

x + 3

x

= 4

= 4

= 1

Eq. (1)

Let y = 3.

 

Check that (1, 3,-4) satisfies all three of the original equations. The solution set is {(1, 3, -4)}.

Hint

In the previous example we chose to eliminate y first. Try solving Example 2 by first eliminating z. Write z = 2 - 2y and then substitute 2 - 2y for z in Eqs. (1) and (2).

In solving a system in three variables it is essential to keep your work organized and neat. Writing short notes that explain your steps (as was done in the examples) will allow you to go back and check your work.

 
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