Simultaneous Equations

Simultaneous equations are two or more equations with the same variables that are all true at the same time. In this topic you will learn to solve 2×2 linear systems by substitution, elimination, and by using matrices.

  • Substitution method
  • Elimination by addition/subtraction
  • Solving using matrices
  • Special cases and checking

Lessons for Simultaneous Equations

Substitution Method

Idea of substitutionMake one variable the subject in one equation, then substitute that expression into the other equation.Example:y = 2x …

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Elimination Method - Addition and Subtraction

Idea of eliminationWe eliminate one variable by adding or subtracting the equations, then solve for the remaining variable.When to addIf …

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Elimination - Multiplying First

When coefficients are not equalMultiply one or both equations so that one variable has the same coefficient, then add or …

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Matrix Method for 2×2 Systems

Writing as a matrix equationSystem:ax + by = ecx + dy = fBecomes: [[a, b], [c, d]] [[x], [y]] = …

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Special Cases and Checking

No solutionIf after elimination you get 0 = 5 or determinant = 0, the lines are parallel. No solution.Example: x …

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Exercises for Simultaneous Equations

Solve by substitution

Solve by substitution:y = x + 32x + y = 9x = 2y - 13x + 4y = 11y = …

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Solve by adding or subtracting

Solve by elimination:3x + y = 102x - y = 54x + 3y = 174x - 2y = 75x + …

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Multiply then eliminate

Solve by elimination:3x + 2y = 122x + 3y = 134x - y = 73x + 2y = 185x + …

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Solve using matrices

Solve using the matrix method:x + y = 7x - y = 12x + y = 5x + 3y = …

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Mixed problems - choose your method

Solve using any method. State if there is a unique solution, no solution, or infinite solutions:y = 2x - 13x …

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