Skip to content

Physics Playground

Simulations How we teach Pricing

Language

Log in Start learning free

Loading…

Physics Playground

Built on open, peer-reviewed sources: OpenStax, The Feynman Lectures on Physics, HyperPhysics, NASA and NIST. See our sources.

Learn

Physics course
Math for Physics
Start with the math toolkit
Simulations
Glossary
Exercises
Past exams
Library
How we teach

Company

Pricing
Terms
Privacy
Contact

© 2026 Physics Playground. Made for the curious.Language

Exercises Systems of equations

Example 2 from Algebra and Trigonometry, 11.8 Solving Systems with Cramer's Rule

Using Cramer’s Rule to Solve a 2 × 2 System

Solve the following $2 \times 2$ system using Cramer’s Rule.

\begin{matrix} 12 x + 3 y = 15 \\ 2 x - 3 y = 13 \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
Solve for $x .$

$x = \frac{D_{x}}{D} = \frac{| \begin{array}{r} 15 & 3 \\ 13 & - 3 \end{array} |}{| \begin{array}{r} 12 & 3 \\ 2 & - 3 \end{array} |} = \frac{- 45 - 39}{- 36 - 6} = \frac{- 84}{- 42} = 2$
Solve for $y .$

$y = \frac{D_{y}}{D} = \frac{| \begin{array}{r} 12 & 15 \\ 2 & 13 \end{array} |}{| \begin{array}{r} 12 & 3 \\ 2 & - 3 \end{array} |} = \frac{156 - 30}{- 36 - 6} = - \frac{126}{42} = −3$
The solution is $(2, −3) .$

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 2 × 2 system of equations.

\begin{array}{l} x + 2 y = −11 \\ - 2 x + y = −13 \end{array}

How did it go?

PreviousFinding the Determinant of a 2 × 2 Matrix Next Finding the Determinant of a 3 × 3 Matrix

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.