Simultaneous Equations
Year 9 📊 Algebra Solve simultaneous equations by substitution and elimination.
🔗 What are Simultaneous Equations?
Two equations with two unknowns that must both be satisfied at the same time. Geometrically, the solution is where two lines intersect.
System: $2x + y = 7$ and $x - y = 2$. We need values of $x$ and $y$ that satisfy both equations simultaneously.
➕ Elimination Method
Add or subtract equations to eliminate one variable, then solve for the other.
- Make the coefficient of one variable equal in both equations (multiply if needed).
- Add or subtract the equations to eliminate that variable.
- Solve for the remaining variable.
- Substitute back to find the eliminated variable.
Example: $2x + y = 7$ and $x - y = 2$. Add: $3x = 9 \Rightarrow x = 3$. Sub back: $3 - y = 2 \Rightarrow y = 1$. Answer: $(3, 1)$
🔄 Substitution Method
Rearrange one equation to express one variable in terms of the other, then substitute into the second equation.
Example: $y = 2x - 1$ and $3x + 2y = 12$. Substitute $y$: $3x + 2(2x-1) = 12$ → $7x - 2 = 12$ → $x = 2$, then $y = 3$.
Substitution works best when one equation already has a variable isolated (e.g. $y = \ldots$).
Ready to test yourself? Click the Quiz tab above to answer questions on this topic!
Interactive Demonstration — Simultaneous Equations
Adjust the two equations and see them intersect
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🔗 Simultaneous Equation Solver
Solve a₁x + b₁y = c₁ and a₂x + b₂y = c₂.