Linear Equations & Graphs
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Lesson overview
Solving simultaneous linear equations and interpreting their graphs.
Go to lesson quiz →Exam guidance · WAEC WASSCE Paper 1 + Paper 2 Lens
Core idea (Linear & Quadratic Equations, Graphs & Inequalities): Rearrange to standard form ax² + bx + c = 0, check the discriminant b² − 4ac for root count, and verify sign flips when dividing inequalities by negatives.
Pacing
Objective speed on Paper 1 + structured step-by-step method marks on Paper 2 Theory/Practical.
How WAEC marks this
Examiners award method marks (M1) for stating the formula/definition and accuracy marks (A1) ONLY when SI units are written.
Watch out for
Writing bare numbers without working or omitting SI units loses up to 50% of Paper 2 marks even when the final value is right.
Lesson content
Overview — Linear Equations & Graphs
Solving simultaneous linear equations and interpreting their graphs. This lesson is at the beginner level — by the end, you'll be able to recognise this question type, apply a reliable method, and avoid the traps that cost marks.
Key Concept
A linear equation in one variable has the form ax + b = c, solved by isolating x. Further Mathematics extends this immediately to SIMULTANEOUS linear equations (two equations, two unknowns), solved by substitution or elimination, and interprets their graphs as intersecting straight lines.
Why It's Tested This Way
Further Maths tests simultaneous equations because they recur throughout the syllabus (in matrices, vectors, and coordinate geometry) as the underlying technique for finding where two conditions are both satisfied at once.
Method
To solve simultaneous equations by elimination, multiply one or both equations so a variable's coefficients match, then add or subtract the equations to eliminate that variable, solving for the other; back-substitute to find the first variable.
Worked Example
Worked example: Solve 2x + y = 8 and x − y = 1. Adding the two equations eliminates y: 3x = 9, so x = 3. Substituting into x − y = 1 gives y = 2.
Second Worked Example
Second worked example: Graphically, the solution (x = 3, y = 2) is exactly the POINT where the two lines 2x + y = 8 and x − y = 1 intersect on a coordinate plane — the algebraic and graphical methods must always agree.
Common Traps
Common trap: adding or subtracting the equations in a way that eliminates the WRONG variable, or forgetting to apply the same multiplication to BOTH sides of an equation before combining.
Lesson Summary
Remember: (1) identify what is being asked, (2) apply the WAEC strategy, (3) check against the traps. Complete the 10-question quiz below to unlock the next lesson — you need at least 60% to pass.
Mastery Check · Lesson Quiz · pass 60%
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