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Exercises Systems of equations

Example 4 from Algebra and Trigonometry, 11.8 Solving Systems with Cramer's Rule

Solving a 3 × 3 System Using Cramer’s Rule

Find the solution to the given 3 × 3 system using Cramer’s Rule.

\begin{matrix} x + y - z = 6 \\ 3 x - 2 y + z = −5 \\ x + 3 y - 2 z = 14 \end{matrix}

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
Use Cramer’s Rule.

$D = | \begin{matrix} 1 & 1 & - 1 \\ 3 & - 2 & 1 \\ 1 & 3 & - 2 \end{matrix} |, D_{x} = | \begin{matrix} 6 & 1 & - 1 \\ - 5 & - 2 & 1 \\ 14 & 3 & - 2 \end{matrix} |, D_{y} = | \begin{matrix} 1 & 6 & - 1 \\ 3 & - 5 & 1 \\ 1 & 14 & - 2 \end{matrix} |, D_{z} = | \begin{matrix} 1 & 1 & 6 \\ 3 & - 2 & - 5 \\ 1 & 3 & 14 \end{matrix} |$

Then,

$\begin{array}{l} x = \frac{D_{x}}{D} = \frac{- 3}{- 3} = 1 \\ y = \frac{D_{y}}{D} = \frac{- 9}{- 3} = 3 \\ z = \frac{D_{z}}{D} = \frac{6}{- 3} = - 2 \end{array}$
The solution is $(1, 3, −2) .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Use Cramer’s Rule to solve the 3 × 3 matrix.

\begin{array}{r} x - 3 y + 7 z = 13 \\ x + y + z = 1 \\ x - 2 y + 3 z = 4 \end{array}

How did it go?

PreviousFinding the Determinant of a 3 × 3 Matrix Next Using Cramer’s Rule to Solve an Inconsistent System

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 11.8, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.