Find values satisfying two equations together

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Find values satisfying two equations together

A simultaneous solution must satisfy both equations with the same values of the variables. Elimination adds or subtracts equations to remove one unknown; substitution replaces one variable using a relationship from the other equation.

Choose multipliers so the coefficients of one variable match or are opposites. Apply a multiplier to every term of an equation. After finding the first variable, substitute to find the second and check the pair in both original equations.

On a graph, a simultaneous solution is an intersection. Parallel distinct lines have no shared point; identical line equations describe infinitely many shared points. A graphical reading normally gives an estimate, while exact algebra can avoid plotting error.

Worked example: Solve x + y = 7 and x − y = 1.

Add the equations to eliminate y: 2x = 8.

So x = 4.

Substitute into x + y = 7 to get y = 3.

Answer: (x, y) = (4, 3).

Check your understanding: Why is (4, 2) not a solution?

Answer: It fails both equations.

Why: 4 + 2 = 6, not 7, and 4 − 2 = 2, not 1. A simultaneous solution must satisfy both original equations.