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, 11.3 Systems of Nonlinear Equations and Inequalities: Two Variables

Finding the Intersection of a Circle and a Line by Substitution

Find the intersection of the given circle and the given line by substitution.

\begin{array}{l} x^{2} + y^{2} = 5 \\ y = 3 x −5 \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
One of the equations has already been solved for $y .$ We will substitute $y = 3 x −5$ into the equation for the circle.

$\begin{matrix} x^{2} + {(3 x −5)}^{2} = 5 \\ x^{2} + 9 x^{2} −30 x + 25 = 5 \\ 10 x^{2} −30 x + 20 = 0 \end{matrix}$
Now, we factor and solve for $x .$

$\begin{array}{l} 10 (x^{2} - 3 x + 2) = 0 \\ 10 (x - 2) (x - 1) = 0 \\ x = 2 \\ x = 1 \end{array}$
Substitute the two x-values into the original linear equation to solve for $y .$

$\begin{array}{l} y = 3 (2) −5 \\ = 1 \\ y = 3 (1) −5 \\ = −2 \end{array}$
The line intersects the circle at $(2, 1)$ and $(1, −2),$ which can be verified by substituting these $(x, y)$ values into both of the original equations. See Figure 5.

Figure 5

Your turn

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

Solve the system of nonlinear equations.

\begin{matrix} x^{2} + y^{2} & = & 10 \\ x - 3 y & = & -10 \end{matrix}

How did it go?

PreviousSolving a System of Nonlinear Equations Representing a Parabola and a Line Next Solving a System of Nonlinear Equations Representing a Circle and an Ellipse

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 11.3, 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.