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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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).
When two distinct lines cross, their crossing point is the only pair of values that makes both equations true.
Includes adapted readings credited to the academy collection, alongside original practice questions. Individual source titles are listed with the readings. The source licenses are unknown. Rewritten readings were written for this dataset.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
16 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.