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Simultaneous equations

A simultaneous solution must satisfy both equations at once. Elimination is efficient when coefficients match or can be made to match; substitution is convenient when one unknown is already isolated.

What this guide covers

Simultaneous linear equations in two unknowns by elimination, substitution and graphical interpretation. This guide teaches the two algebraic methods and explains the intersection meaning.

Syllabus reference: 2026 Mathematics 4052, N7, page 7. These are Mentora search groupings, not new official syllabus categories.

The content in this guide was also checked against 2027 G3 Mathematics K310, N7, page 7 on 2026-09-12. This is subject-content alignment, not a claim of complete syllabus coverage.

Worked example

Solve 2x + y = 11 and x − y = 1 by elimination.

Add the equations: 3x = 12, so x = 4. Substitute into x − y = 1 to get y = 3. Check both: 2(4) + 3 = 11 and 4 − 3 = 1.

Substitution on the same pair

Now solve 2x + y = 11 and x − y = 1 by substitution.

From x − y = 1, write y = x − 1. Substitute the whole expression: 2x + (x − 1) = 11. Then 3x = 12, x = 4 and y = 3. Both methods give the same intersection (4, 3). Elimination is shorter here because +y and −y already cancel; substitution is equally valid.

Try three practice checks

Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.

Check 1

Solve x + y = 9 and x − y = 3.

Show worked answer

Add to give 2x = 12, so x = 6 and y = 3. Both 6 + 3 = 9 and 6 − 3 = 3 check.

Check 2

Solve y = 2x + 1 and 3x + y = 16.

Show worked answer

Substitute: 3x + 2x + 1 = 16, so x = 3 and y = 7. Check 9 + 7 = 16.

Check 3

Two notebooks and one pen cost $11. One notebook and two pens cost $10. Find each price.

Show worked answer

Let n and p be prices in dollars. 2n + p = 11 and n + 2p = 10. Double the first and subtract the second: 3n = 12, so n = 4 and p = 3. Check 2(4) + 3 = 11 and 4 + 2(3) = 10.

Mistakes to watch for

  • Multiplying an equation means multiplying every term and the right-hand side.
  • Subtract the entire second equation, including negative terms.
  • One equation alone is not enough to verify a pair of unknowns; substitute into both.

If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.

Useful foundations to check

These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.

Applications and next topics

Graph each equation as a straight line: the solution is their intersection. Parallel distinct lines have no common solution; coincident lines have infinitely many. Use equations to model two related quantities before solving.

Choose a useful next resource

To understand a recurring error rather than relearn the whole skill: Connect solutions to graph intersections.

Explore the foundations connected to simultaneous equations

Sources and review

Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-12; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.