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Exercises Systems of equations

Example 3 from Algebra and Trigonometry, 11.6 Solving Systems with Gaussian Elimination

Solving a 2×2 System by Gaussian Elimination

Solve the given system by Gaussian elimination.

\begin{array}{l} 2 x + 3 y = 6 \\ x - y = \frac{1}{2} \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
First, we write this as an augmented matrix.

$[\begin{array}{r} 2 & 3 \\ 1 & −1 \end{array} | \begin{array}{r} 6 \\ \frac{1}{2} \end{array}]$
We want a 1 in row 1, column 1. This can be accomplished by interchanging row 1 and row 2.

$R_{1} \leftrightarrow R_{2} \to [\begin{array}{r} 1 & −1 \\ 2 & 3 \end{array} | \begin{array}{r} \frac{1}{2} \\ 6 \end{array}]$
We now have a 1 as the first entry in row 1, column 1. Now let’s obtain a 0 in row 2, column 1. This can be accomplished by multiplying row 1 by $−2,$ and then adding the result to row 2.
$−2 R_{1} + R_{2} = R_{2} \to [\begin{array}{r} 1 & −1 \\ 0 & 5 \end{array} | \begin{array}{r} \frac{1}{2} \\ 5 \end{array}]$
We only have one more step, to multiply row 2 by $\frac{1}{5} .$

$\frac{1}{5} R_{2} = R_{2} \to [\begin{array}{r} 1 & −1 \\ 0 & 1 \end{array} | \begin{matrix} \frac{1}{2} \\ 1 \end{matrix}]$
Use back-substitution. The second row of the matrix represents $y = 1.$ Back-substitute $y = 1$ into the first equation.

$\begin{array}{l} x - (1) = \frac{1}{2} \\ x = \frac{3}{2} \end{array}$
The solution is the point $(\frac{3}{2}, 1) .$

Your turn

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

Solve the given system by Gaussian elimination.

\begin{array}{l} 4 x + 3 y = 11 \\ x−3y=−1 \end{array}

How did it go?

PreviousWriting a System of Equations from an Augmented Matrix Form Next Using Gaussian Elimination to Solve a System of Equations

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.