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Exercises Systems of equations

Example 4 from Algebra and Trigonometry, 11.6 Solving Systems with Gaussian Elimination

Using Gaussian Elimination to Solve a System of Equations

Use Gaussian elimination to solve the given $2 \times 2$ system of equations.

\begin{array}{l} 2 x + y = 1 \\ 4 x + 2 y = 6 \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 system as an augmented matrix.

$[\begin{array}{l} 2 & 1 \\ 4 & 2 \end{array} | \begin{array}{l} 1 \\ 6 \end{array}]$
Obtain a 1 in row 1, column 1. This can be accomplished by multiplying the first row by $\frac{1}{2} .$

$\frac{1}{2} R_{1} = R_{1} \to [\begin{matrix} 1 & \frac{1}{2} \\ 4 & 2 \end{matrix} | \begin{matrix} \frac{1}{2} \\ 6 \end{matrix}]$
Next, we want a 0 in row 2, column 1. Multiply row 1 by $−4$ and add row 1 to row 2.
$−4 R_{1} + R_{2} = R_{2} \to [\begin{matrix} 1 & \frac{1}{2} \\ 0 & 0 \end{matrix} | \begin{matrix} \frac{1}{2} \\ 4 \end{matrix}]$
The second row represents the equation $0 = 4.$ Therefore, the system is inconsistent and has no solution.

How did it go?

PreviousSolving a 2×2 System by Gaussian Elimination Next Solving a Dependent System

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.