---
title: "Simultaneous equations"
description: "Understand that a simultaneous solution is the pair of values at which two linear graphs meet, and verify that pair by substituting into both equations."
canonical: https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz
source: https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz.md
retrieved: 2026-10-19
---

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# Simultaneous equations

Understand that a simultaneous solution is the pair of values at which two linear graphs meet, and verify that pair by substituting into both equations.

Subject: Mathematics · Area: Algebra · Ages 13 to 14
Page: https://lightmysky.com/learn/mathematics/simultaneous-equations-mt_mqgu72aCMz

## Ready when they can

- Explain that the intersection of two lines represents values satisfying both equations
- Find the approximate solution to a pair of simultaneous equations by reading a graph
- Verify a proposed solution by substituting into both equations

## Lesson: Where two lines meet

Solving two equations simultaneously means finding all pairs of x and y values that make both true. Here we focus on distinct lines that cross once, like y = x + 2 and y = 3x - 4. Their crossing gives one solution. Distinct parallel lines have no solution; equations drawing the same line have infinitely many.

If you plot both equations as lines on the same graph, each line shows every point that fits its own equation. When the two lines cross at one point, that crossing point is the only pair of values that satisfies both equations together.

**Example.** Zara graphs y = x + 1 and y = -x + 5 on the same axes. The lines cross at the point (2, 3). She checks: for the first equation, 3 = 2 + 1 is true. For the second, 3 = -2 + 5 is true. So x = 2, y = 3 solves both equations.

*(drawing: For two distinct lines that cross: draw both lines, find their crossing point, then check its coordinates in both equations.)*

To check a proposed point, substitute its x and y values into both equations. If both equations are true, the point solves the system. If one fails, it is not a solution.

Coordinates from a graph use the order (x, y). Read the horizontal value first and the vertical value second. For an estimate, keep each coordinate close to the crossing shown on the graph.

Substitution can check a graph estimate. At x = 7, y = 2x - 5 gives y = 9 and y = x + 2 also gives y = 9. Matching outputs support the point (7, 9).

**Recap.** When two distinct lines cross, their crossing point is the only pair of values that makes both equations true.

## Practice

16 questions on this page, each with its working shown.

## Needs first

- [Plotting Linear Graphs](https://lightmysky.com/learn/mathematics/plotting-linear-graphs-mt_-3udyo6VyB)
- [Solving linear equations](https://lightmysky.com/learn/mathematics/solving-linear-equations-mt_QhFEDyIwSO)
- [Equations with Two Unknowns](https://lightmysky.com/learn/mathematics/equations-with-two-unknowns-mt_YFS7JFk64p)

## Opens up

- [Simultaneous Equations with One Quadratic](https://lightmysky.com/learn/mathematics/simultaneous-equations-with-one-quadratic-mt_D89ql2sJvc)
- [Connected Particles and Pulleys](https://lightmysky.com/learn/mathematics/connected-particles-and-pulleys-mt_pZ6zV0RFdC)
- [Systems of linear equations and their solution sets](https://lightmysky.com/learn/mathematics/systems-of-linear-equations-and-their-solution-sets-mt_sQL6JfR7Te)