Exponential Functions
Linera Equations
Simple Trinomials as Products of Binomials
Laws of Exponents and Dividing Monomials
Solving Equations
Multiplying Polynomials
Multiplying and Dividing Rational Expressions
Solving Systems of Linear Inequalities
Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50

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The techniques for solving a system of linear equations in three variables are similar to those used on systems of linear equations in two variables. We eliminate variables by either substitution or addition.

Example 1

A linear system with a single solution

Solve the system:

(1) x + y - z = -1

(2) 2x - 2y + 3z = 8

(3) 2x - y + 2z = 9


We can eliminate z from Eqs. (1) and (2) by multiplying Eq. (1) by 3 and adding it to Eq. (2):

  3x + 3y - 3z = -3 Eq. (1) multiplied by 1
  2x - 2y + 3z = 8 Eq.(2)
(4) 5x + y   = 5

Now we must eliminate the same variable, z, from another pair of equations. Eliminate z from (1) and (3):

  2x + 2y - 2z = -2 Eq. (1) multiplied by 2
  2x - y + 2z = 9   Eq. (3)
(5) 4x + y   = 7  

Equations (4) and (5) give us a system with two variables. We now solve this system. Eliminate y by multiplying Eq. (5) by -1 and adding the equations:

5x + y = 5 Eq. (4)
-4x - y = -7 Eq. (5) multiplied by -1
x = -2  

Now that we have x, we can replace x by -2 in Eq. (5) to find y:

4x + y

4(-2) + y

 -8 + y


= 7

= 7

= 7

= 15

Now replace x by -2 and y by 15 in Eq. (1) to find z:

x + y - z

-2 + 15 - z

13 - z



= -1

= -1

= -1

= -14

= 14

 Check that (-2, 15, 14) satisfies all three of the original equations. The solution set is {(-2, 15, 14)}. 

Helpful Hint

Note that we could have chosen to eliminate x, y, or z first in Example 1.You should solve this same system by eliminating x first and then by eliminating y first. To eliminate x first, multiply the first equation by -2 and add it with the second and third equations. In the next example we use a combination of addition and substitution.