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Simultaneous Equations - Edexcel GCSE Higher Maths

Typically 3-5 marks per paper, up to ≈9 marks across a series · Spec 1MA1

Simultaneous equations ask you to find values that satisfy two equations at once. On Edexcel's 1MA1 Higher tier they appear in two forms: a pair of linear equations (spec ref A19), usually worth 3-4 marks in the middle of a paper, and the Higher-only pairing of one linear with one quadratic equation (including a circle x2+y2=r2x^2 + y^2 = r^2 with a line), worth around 5 marks late in the paper.

Edexcel can set them on any of the three papers, and Paper 1 is non-calculator, so you must be comfortable eliminating by hand and giving exact answers: mark schemes regularly have fractional solutions like x=12x = \frac{1}{2} precisely to catch students who assume answers are whole numbers.

The method marks (M marks) are earned for a correct elimination or substitution process even if arithmetic slips later, and the accuracy marks (A marks) need both unknowns: examiner reports repeatedly note candidates who find xx and never return for yy. Practise the full loop: eliminate, solve, substitute back, check in the other original equation.

Worked example

Solve the simultaneous equations 5x+2y=115x + 2y = 11 3x−4y=43x - 4y = 4
[4 marks]
  1. Multiply the first equation by 2 so the yy coefficients match in size: 10x+4y=2210x + 4y = 22.
    M1: scaling one equation to equalise coefficients (every term must be multiplied, including the 11)
  2. The yy terms are +4y+4y and −4y-4y: opposite signs, so add the equations: 13x=2613x = 26, giving x=2x = 2.
    A1: correct first unknown from a correct elimination
  3. Substitute x=2x = 2 into 5x+2y=115x + 2y = 11: 10+2y=1110 + 2y = 11, so 2y=12y = 1 and y=12y = \frac{1}{2}.
    M1: substituting their xx back into an original equation
  4. State both values: x=2x = 2, y=12y = \frac{1}{2}. Check in the other equation: 3(2)−4(12)=6−2=43(2) - 4(\frac{1}{2}) = 6 - 2 = 4 ✓
    A1: both values correct; the fractional yy is deliberate, this is a non-calculator skill

Practice questions

These are original questions in Edexcel 1MA1 Higher style. Simultaneous equations can appear on any of the three papers, and Paper 1 is non-calculator, so try the linear pairs without a calculator and give exact fractions (you can type answers like 1/2). The harder questions mirror the Higher-only linear-plus-quadratic style.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [3 marks]
    Solve the simultaneous equations x+y=10x + y = 10 x−y=4x - y = 4
  2. Question 2 Warm-up [3 marks]
    Solve the simultaneous equations 4x+y=144x + y = 14 2x+y=82x + y = 8
  3. Question 3 Warm-up [4 marks]
    Solve the simultaneous equations 5x+2y=205x + 2y = 20 2x+y=82x + y = 8
  4. Question 4 Warm-up [3 marks]
    A rectangle's length is 3 cm more than its width. Its perimeter is 26 cm. Find the width and length of the rectangle, in cm.
  5. Question 5 Exam pace [2 marks]
    Given 3x−2y=123x - 2y = 12, express yy in terms of xx.
  6. Question 6 Stretch [3 marks]
    Solve the simultaneous equations y=x2y = x^2 and y=x+6y = x + 6, and give the positive value of xx.

Common mistakes examiners see

  • Adding the equations when the matched coefficients have the same sign (or subtracting when they differ), so the variable is not eliminated.

    Say it out loud before you combine: Same Signs Subtract, opposite signs add. Then write the combined equation on its own line; examiners award the M mark for a correct elimination even if later arithmetic slips.

  • Multiplying an equation but forgetting the constant term, turning 3x+2y=193x + 2y = 19 into 6x+4y=196x + 4y = 19.

    Multiply every term on both sides. A one-second check: the new equation should still be true for the same xx and yy.

  • Finding the first unknown and stopping, or finding xx and yy but never stating them together.

    'Solve' means both values. On Edexcel mark schemes the final A mark is for the pair, so write x=…,y=…x = \ldots, y = \ldots as your last line every time.

  • Substituting back into the equation you just rearranged, which hides any earlier error.

    Always check your pair in the other original equation; it costs ten seconds and catches sign slips before the examiner does.

  • Assuming the answers must be whole numbers and abandoning correct working when a fraction appears on Paper 1.

    Edexcel deliberately sets non-integer solutions on the non-calculator paper. Keep values as exact fractions rather than forcing a decimal.

Frequently asked questions

How many marks are simultaneous equations worth on Edexcel Higher?
A linear pair is usually 3-4 marks, and the Higher-only linear-plus-quadratic question is typically 5 marks. Across a full series (three papers, 240 marks) expect roughly 3-9 marks, so it is one of the more reliable algebra topics to bank.
Are simultaneous equations on the non-calculator paper?
They can be on any paper. Edexcel regularly puts a linear pair on Paper 1 (non-calculator), often with a fractional answer, so practise eliminating and substituting by hand.
Do I need the quadratic version for Higher tier?
Yes. Higher tier includes solving one linear and one quadratic equation simultaneously (including a line with the circle x2+y2=r2x^2 + y^2 = r^2), usually by substituting the linear equation into the quadratic. Foundation tier only requires two linear equations.

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