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Exercises Systems of equations

Example 9 from Algebra and Trigonometry, 11.6 Solving Systems with Gaussian Elimination

Solving Systems of Equations with Matrices Using a Calculator

Solve the system of equations.

\begin{array}{r} 5 x + 3 y + 9 z = −1 \\ −2 x + 3 y - z = −2 \\ - x −4 y + 5 z = 1 \end{array}

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
Write the augmented matrix for the system of equations.

$[\begin{array}{r} 5 & 3 & 9 \\ −2 & 3 & −1 \\ −1 & −4 & 5 \end{array} | \begin{array}{r} −1 \\ −2 \\ −1 \end{array}]$
On the matrix page of the calculator, enter the augmented matrix above as the matrix variable $[A] .$

$[A] = [\begin{array}{r} 5 & 3 & 9 & −1 \\ −2 & 3 & −1 & −2 \\ −1 & −4 & 5 & 1 \end{array}]$
Use the ref( function in the calculator, calling up the matrix variable $[A] .$

$ref ([A])$
Evaluate.

$\begin{array}{l} [\begin{array}{r} 1 & \frac{3}{5} & \frac{9}{5} & \frac{1}{5} \\ 0 & 1 & \frac{13}{21} & - \frac{4}{7} \\ 0 & 0 & 1 & - \frac{24}{187} \end{array}] \to \begin{array}{l} x + \frac{3}{5} y + \frac{9}{5} z = - \frac{1}{5} \\ y + \frac{13}{21} z = - \frac{4}{7} \\ z = - \frac{24}{187} \end{array} \end{array}$
Using back-substitution, the solution is $(\frac{61}{187}, - \frac{92}{187}, - \frac{24}{187}) .$

How did it go?

PreviousSolving a Dependent System of Linear Equations Using Matrices Next Applying 2 × 2 Matrices to Finance

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 11.6, 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.