Point of Intersection of Two Lines

Finding where two lines meet is a foundational step that appears inside many longer 0606 questions, perpendicular bisectors, triangle problems, tangents. The intersection is the point satisfying both equations, so you solve them simultaneously.

Find the coordinates of the point of intersection of the lines $2x + 3y = 12$ and $x - y = 1$. [3]

The working

Step 1, make a variable the subject of the simpler equation. From $x - y = 1$: $x = y + 1 \quad \text{(M1)}$

Step 2, substitute into the other line: $2(y + 1) + 3y = 12 \;\Rightarrow\; 2y + 2 + 3y = 12 \;\Rightarrow\; 5y = 10 \;\Rightarrow\; y = 2 \quad \text{(A1)}$

Step 3, back-substitute for $x$ using $x = y + 1$: $x = 2 + 1 = 3$

The lines meet at $(3, 2)$. (A1)

A quick check: $2(3) + 3(2) = 6 + 6 = 12$ ✓ and $3 - 2 = 1$ ✓, both satisfied.

Where the marks are won and lost

• Substitute the rearranged expression into the other equation, not back into the one it came from (that gives $0 = 0$).
• Give both coordinates. The intersection is a point $(3, 2)$; a single value is incomplete.
• Check by substituting into both original equations, cheap insurance against an arithmetic slip.

Common mistakes

• Substituting $x = y + 1$ back into $x - y = 1$ (circular, gives nothing).
• Solving for one variable and forgetting the other.
• Sign slips expanding $2(y + 1)$.

Full method: Equation of a Line notes. See also Points of Intersection. Topic home: Straight-Line Graphs pillar.