Simultaneous Equations · Hook

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Ever wondered how to work out two prices from two receipts?

Kemi paid ₦950 for 2 exercise books and 1 biro. Musa paid ₦2,050 for 4 exercise books and 3 biros. Neither receipt shows the price of one book — but together, the two receipts give it away. That is the power of simultaneous equations.

Revision notes

📒 Simultaneous Equations: lesson notes

Solve two equations with two unknowns by elimination or substitution, and use them to crack price and ticket problems.

Kemi paid ₦950 for 2 exercise books and 1 biro. Musa paid ₦2,050 for 4 exercise books and 3 biros. Neither receipt shows the price of one book — but together, the two receipts give it away. That is the power of simultaneous equations.

Two equations, two unknowns

One equation like x + y = 10 has many answers (1 and 9, 4 and 6, …). A second equation, x − y = 2, narrows it down to one pair that fits both: x = 6, y = 4.

'Simultaneous' means both equations are true at the same time. Always check your answer in both equations.

🔑 Key idea: The solution is the one pair of values that makes BOTH equations true.

Elimination

Make the coefficients of one letter the same size, then add or subtract to remove it. Same signs → subtract; opposite signs → add.

For 3x + 2y = 16 and x + 2y = 8: subtracting gives 2x = 8, so x = 4. Then 4 + 2y = 8, so y = 2.

If the coefficients don't match, multiply one or both equations first: to match 2y and 3y, multiply by 3 and by 2 to get 6y in both.

🔑 Key idea: Same signs → subtract. Opposite signs → add.

Substitution

If one equation already says y = …, put that expression in place of y in the other equation.

For y = 2x + 1 and 3x + y = 11: 3x + (2x + 1) = 11, so 5x = 10 and x = 2. Then y = 2(2) + 1 = 5.

In word problems, first say what each letter stands for, then write one equation for each fact.

🔑 Key idea: Replace a letter with what it equals, then solve the one-letter equation.

Worked example

Solve the simultaneous equations 3x + 4y = 23 and 4x + 4y = 28. Find x.

  1. Step 1: Label the equations

    3x + 4y = 23 …(1)

    4x + 4y = 28 …(2)

  2. Step 2: (1) − (2) eliminates y

    −x = −5

    x = 5

    Why? 4y − 4y = 0

  3. Step 3: Substitute into (1)

    3(5) + 4y = 23

    4y = 23 − 15 = 8

    y = 2

  4. Step 4: Check in (2)

    4(5) + 4(2) = 28 ✓

Answer: x = 5

Summary

  • Simultaneous equations are both true at the same time — find the pair that fits both.
  • Elimination: match one letter's coefficients, then add (opposite signs) or subtract (same signs).
  • Substitution: replace y with its expression, and multiply every term in the bracket.
  • In word problems, define your letters first, then check your answer makes sense.

Ready to check you've got it? Scroll up and work through the lesson: the interactive visuals, practice questions and mastery check are all there.