Solving Simultaneous Equations: Substitution & Elimination
Year 10 Essential Mathematics Understanding two methods for solving systems of equations
What Are Simultaneous Equations?
Two (or more) equations with the same unknowns We need to find values that satisfy ALL equations at the same time Example: x + y = 10 and x - y = 2 Looking for one pair of values (x, y) that makes both equations true
Why Do We Use Simultaneous Equations?
Cost and budgeting problems Distance, speed, and time calculations Number puzzles and word problems Business and finance situations Any scenario with two relationships happening at once
Key Insight
When two relationships exist at the same time, we can model them using simultaneous equations to find the exact solution.
Method 1: Substitution
Make one variable the subject of one equation Replace (substitute) it into the other equation Solve the resulting single-variable equation Substitute back to find the second variable Perfect when one equation is easy to rearrange
Substitution Example
Given: y = 2x + 1 and x + y = 7 Step 1: Substitute y = 2x + 1 into x + y = 7 Step 2: x + (2x + 1) = 7 → 3x + 1 = 7 → x = 2 Step 3: Substitute back: y = 2(2) + 1 = 5 Solution: x = 2, y = 5
Method 2: Elimination
Line up equations in standard form Add or subtract to eliminate one variable Make coefficients opposites if needed Solve for the remaining variable Substitute back to find the second variable Best when coefficients are already similar
Elimination Example
Given: 2x + y = 9 and x + y = 6 Step 1: Subtract second equation from first (2x + y) - (x + y) = 9 - 6 Step 2: Simplify: x = 3 Step 3: Substitute back: 3 + y = 6, so y = 3 Solution: x = 3, y = 3
When to Use Each Method
{"left":"Use Substitution when:\nA variable is already isolated\nOne equation is easy to rearrange\nCoefficients are fractions or decimals","right":"Use Elimination when:\nCoefficients are the same or similar\nEquations are in standard form\nVariables have integer coefficients"}
Key Reminders & Next Steps
Always show all working clearly Substitute back to check your answer Write final answer as coordinate pairs: (x, y) Check: does your solution satisfy both original equations? Practice with different types of problems Next lesson: Graphical method and applications