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§ Algebra

Systems of Equations

CCSS.8.EECCSS.HSA.REI3 min read

Systems of equations appear in Year 9 GCSE preparation when students must find where two lines intersect on a coordinate plane. These simultaneous equations require methodical substitution or elimination techniques to determine the unique solution pair (x, y).

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§ 01

Why it matters

Systems of equations solve countless real-world problems across business, engineering, and everyday scenarios. A chippy owner might use them to determine how many portions of fish (£4.50 each) and chips (£2.20 each) were sold if total sales reached £180 from 50 portions. Mobile phone companies use systems to find break-even points between different tariff plans. In GCSE Foundation and Higher papers, these problems often appear as worded scenarios worth 4-6 marks. Engineering students later apply these methods to analyse electrical circuits, calculate material costs in construction projects, and optimise manufacturing processes. The algebraic thinking developed through systems of equations builds essential problem-solving skills students need for A-level Mathematics and beyond.

§ 02

How to solve systems of equations

Systems of Equations

  • Write both equations.
  • Use substitution or elimination to solve for one variable.
  • Substitute back to find the other.
  • Verify in both equations.

Example: x + y = 5, x − y = 1 → x = 3, y = 2.

§ 03

Worked examples

Beginner§ 01

I have two types of coins. Together they are worth £3.00. One type is worth £2.00. How much is the other type worth?

Answer: x = 2, y = 1

  1. Define variables → Let x = value of first coin, y = value of second coin x + y = 3 x = 2 — Translate the word problem into equations.
  2. Label the equations → (1) x + y = 3 (2) x = 2 — Number each equation so we can refer to them.
  3. Solve equation (1) for y → y = 3 − 1x — Isolate y in the simpler equation to use substitution.
  4. Substitute into equation (2) → Substitute y into (2) and solve for x — Replace y in equation (2) with the expression from equation (1), then solve for x.
  5. Find x → x = 2 — Solving gives x = 2.
  6. Substitute x back to find y → In (1): 1·2 + 1·y = 3 → 2 + 1·y = 3 → 1·y = 1 → y = 1 — Plug x = 2 into equation (1) and solve for y.
  7. Write the solution → x = 2, y = 1 — The intersection point of the two lines.
  8. Verify in both equations → (1) 1·2 + 1·1 = 3 = 3 ✓ (2) 1·2 + 0·1 = 2 = 2 ✓ — Substitute the solution into both original equations to confirm.
Easy§ 02

Solve the system: x + y = 4 x − 1y = 0

Answer: x = 2, y = 2

  1. Label the equations → (1) x + y = 4 (2) x − 1y = 0 — Number each equation so we can refer to them.
  2. Solve equation (1) for y → y = 4 − 1x — Isolate y in the simpler equation to use substitution.
  3. Substitute into equation (2) → Substitute y into (2) and solve for x — Replace y in equation (2) with the expression from equation (1), then solve for x.
  4. Find x → x = 2 — Solving gives x = 2.
  5. Substitute x back to find y → In (1): 1·2 + 1·y = 4 → 2 + 1·y = 4 → 1·y = 2 → y = 2 — Plug x = 2 into equation (1) and solve for y.
  6. Write the solution → x = 2, y = 2 — The intersection point of the two lines.
  7. Verify in both equations → (1) 1·2 + 1·2 = 4 = 4 ✓ (2) 1·2 + -1·2 = 0 = 0 ✓ — Substitute the solution into both original equations to confirm.
Medium§ 03

Solve the system: x + 2y = 11 3x + 2y = 21

Answer: x = 5, y = 3

  1. Label the equations → (1) x + 2y = 11 (2) 3x + 2y = 21 — Number each equation so we can refer to them.
  2. Solve equation (1) for y → y = (11 − 1x) / 2 — Isolate y in the simpler equation to use substitution.
  3. Substitute into equation (2) → Substitute y into (2) and solve for x — Replace y in equation (2) with the expression from equation (1), then solve for x.
  4. Find x → x = 5 — Solving gives x = 5.
  5. Substitute x back to find y → In (1): 1·5 + 2·y = 11 → 5 + 2·y = 11 → 2·y = 6 → y = 3 — Plug x = 5 into equation (1) and solve for y.
  6. Write the solution → x = 5, y = 3 — The intersection point of the two lines.
  7. Verify in both equations → (1) 1·5 + 2·3 = 11 = 11 ✓ (2) 3·5 + 2·3 = 21 = 21 ✓ — Substitute the solution into both original equations to confirm.
§ 04

Common mistakes

  • Students often substitute incorrectly, writing x = 3 in both equations instead of checking their work. For x + y = 5 and 2x - y = 1, they might claim x = 3, y = 2, giving 3 + 2 = 5 ✓ but 2(3) - 2 = 4 ≠ 1.
  • When using elimination, students frequently forget to multiply entire equations. Solving x + 2y = 8 and 3x + y = 9, they multiply only the coefficients, getting 3x + 6y = 8 instead of 3x + 6y = 24.
  • Students mix up positive and negative signs during elimination. For x + y = 4 and x - y = 2, they incorrectly add equations as 2x + 0 = 2, giving x = 1 instead of x = 3.
  • Many students stop after finding one variable, forgetting to substitute back. They solve 2x + y = 7 and x = 3 correctly to get x = 3, but submit this as their complete answer without finding y = 1.
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§ 05

Frequently asked questions

Should I use substitution or elimination method?
Use substitution when one equation already isolates a variable (like x = 5) or when coefficients make isolation simple. Choose elimination when coefficients are similar or when multiplying creates easy cancellation. Both methods give identical answers, so practise with your preferred approach first.
How do I check if my solution is correct?
Substitute both x and y values into both original equations. Your solution should satisfy both equations completely. For example, if x = 2 and y = 3, then 2 + 3 should equal 5 in the first equation, and 2(2) - 3 should equal 1 in the second equation.
What if I get fractions or decimals in my answer?
Fractions and decimals are perfectly valid solutions. Always express fractions in simplest form and check your arithmetic carefully. For money problems, round to the nearest penny only at the very end. GCSE mark schemes accept exact fractional answers without decimal conversion.
Why do some systems have no solution or infinite solutions?
Parallel lines (same gradient, different y-intercepts) create no solution, while identical lines create infinite solutions. In Year 9, focus on systems with exactly one solution. These advanced cases typically appear in GCSE Higher papers during coordinate geometry topics.
How do I set up word problems as systems of equations?
Identify two unknown quantities and assign variables. Find two different relationships between these unknowns from the problem text. For example, 'total cost £15' and 'apples cost twice as much as bananas' become x + y = 15 and x = 2y respectively.
§ 06

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