You can use the elimination and substitution methods to solve a system of three equations in three variables by working with the equations in pairs. You will use one of the equations twice. When one point represents the solution of a system of equations in three variables, write it as an ordered triple (x, y, z).
Problem 1 Solving a System Using Elimination
What is the solution of the system? Use elimination. The equations are numbered to make the procedure easy to follow.
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Step 1 Pair the equations to eliminate z. Then you will have two equations in x and y.
Add. Subtract. Think
Which variable do you eliminate first?
Eliminate the variable for which the process requires the fewest steps.
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Step 2 Write the two new equations as a system. Solve for x and y.
Add and solve for y. Substitute y = 3 and solve for x. -
Step 3 Solve for z. Substitute the values of x and y into one of the original equations.
Think
Does it matter which equation you substitute into to find z?
No, you can substitute into any of the original three equations.
- Step 4 Write the solution as an ordered triple. The solution is (3, 3, 1).
✓ Got It?
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What is the solution of the system? Use elimination. Check your answer in all three original equations.
You can apply the method in Problem 1 to most systems of three equations in three variables. You may need to multiply in one, two, or all three equations by one, two, or three nonzero numbers. Your goal is to obtain a system—equivalent to the original system—with coefficients that allow for the easy elimination of variables.
Table of Contents
- 5-1 Polynomial Functions
- 5-2 Polynomials, Linear Factors, and Zeros
- 5-3 Solving Polynomial Equations
- 5-4 Dividing Polynomials
- 5-5 Theorems About Roots of Polynomial Equations
- 5-6 The Fundamental Theorem of Algebra
- 5-7 The Binomial Theorem
- 5-8 Polynomial Models in the Real World
- 5-9 Transforming Polynomial Functions