--- title: "Simultaneous Equations with One Quadratic" description: "Solve a linear and a quadratic equation together by substituting the linear one into the quadratic, and read the answers as the points where a line meets a curve." canonical: https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc source: https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc.md retrieved: 2026-09-02 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Simultaneous Equations with One Quadratic Solve a linear and a quadratic equation together by substituting the linear one into the quadratic, and read the answers as the points where a line meets a curve. Subject: Mathematics · Area: Algebra · Ages 15 to 16 Page: https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc ## Ready when they can - Substitute the linear equation into the quadratic and collect terms to one side - Give the answers as coordinate pairs, not just x values - Say what a single repeated solution means about the line and the curve ## Lesson: Simultaneous Equations with One Quadratic You can now sketch a curve and see where a line would meet it. This stop finds those meeting points exactly. A line and a curve meet where both equations are true at once, so the method is substitution: take y from the linear equation, put it into the quadratic, and solve. The x values you get are the crossing points, and the number of them matches what your sketch would show: two, one, or none. *(drawing: Five moves, ending with the full coordinate pairs.)* The answers must be coordinate pairs, because a solution is a point where the line and the curve meet, and a point needs both an x and a y. Finding x from the quadratic is only half the job: put each x back into the linear equation to get its y. If the quadratic has a repeated root, the line and curve meet at exactly one point, and the line is a tangent to the curve. That is the discriminant-zero case from before, now with a picture. **Example.** Solve y = x squared - 2 and y = x + 4. Substitute the linear into the quadratic: x squared - 2 = x + 4. Collect everything to one side: x squared - x - 6 = 0. Factorise: (x - 3)(x + 2) = 0, so x = 3 or x = -2. Now find each y from the linear equation: when x = 3, y = 7, and when x = -2, y = 2. The line meets the curve at (3, 7) and (-2, 2). **Example.** Now a line that just touches. Solve y = x squared and y = 2x - 1. Substituting: x squared = 2x - 1, so x squared - 2x + 1 = 0. That factorises to (x - 1) squared = 0, so x = 1 is a repeated root. There is only one meeting point. When x = 1, y = 2(1) - 1 = 1, so the line touches the curve at (1, 1). The repeated root is the algebraic version of a tangent. **Example.** And a line that misses. Solve y = x squared + 2 and y = x + 1. Substituting: x squared + 2 = x + 1, so x squared - x + 1 = 0. The discriminant is 1 - 4, which is -3, negative, so this quadratic has no real roots. The line never meets the curve. On a sketch, the curve sits entirely above the line. The discriminant tells you this before you try to factorise, which is useful when the factorisation never comes. **Tip.** Check your points on both equations, not just the one you solved. A point that works in the quadratic but not in the line is not a meeting point. And when a question asks where a line touches a curve, write 'repeated root' or 'discriminant zero' early, because that is the idea the question is testing. **Recap.** Substitute the linear equation into the quadratic, collect to one side and solve, then find each y from the linear equation: two roots give two points, a repeated root gives one tangent point, and a negative discriminant gives none. ## Practice 24 questions on this page, each with its working shown. ## Needs first - [Sketching Quadratic Graphs from Roots and the Turning Point](https://lightmysky.com/learn/mathematics/sketching-quadratic-graphs-from-roots-and-the-turning-point-mt_RcrJIbbN2x) - [Simultaneous Equations](https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz) ## Opens up - [Solving Linear Inequalities and Solution Sets](https://lightmysky.com/learn/mathematics/solving-linear-inequalities-and-solution-sets-mt_QkG3Cw5VgM)