Simultaneous equations (linear) Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Simultaneous equations (linear) questions. See exactly how to solve problems on elimination, adding equations, subtracting equations, eliminating x.

eliminationadding equationssubtracting equationseliminating xnegative constantsnumber problem
GCSE Foundation70 questionsStep-by-step solutions
Question 1
2 markseasy
Solve the simultaneous equations x+y=9x + y = 9 and x−y=3x - y = 3.

Worked solution

  1. Add the equations to eliminate y

    (x+y)+(x−y)=9+3⇒2x=12(x + y) + (x - y) = 9 + 3 \Rightarrow 2x = 12

    The y-terms have opposite signs, so they cancel when you add. That leaves one equation in x only.

  2. Solve for x

    x=122=6x = \frac{12}{2} = 6

    Divide both sides by 2.

  3. Substitute back to find y

    (6)+y=9⇒y=3(6) + y = 9 \Rightarrow y = 3

    Put x=6x = 6 into equation (1) and solve the one-step equation that is left. The solution is x=6x = 6, y=3y = 3.

Answer
x=6, y=3x = 6,\ y = 3
Question 2
2 markseasy
Solve the simultaneous equations 2x+y=112x + y = 11 and x+y=7x + y = 7.

Worked solution

  1. Subtract the equations to eliminate y

    (2x+y)−(x+y)=11−7⇒x=4(2x + y) - (x + y) = 11 - 7 \Rightarrow x = 4

    The y-terms have the same sign, so they cancel when you subtract. That leaves one equation in x only.

  2. Solve for x

    x=41=4x = \frac{4}{1} = 4

    Divide both sides by 1.

  3. Substitute back to find y

    2(4)+y=11⇒y=32(4) + y = 11 \Rightarrow y = 3

    Put x=4x = 4 into equation (1) and solve the one-step equation that is left. The solution is x=4x = 4, y=3y = 3.

Answer
x=4, y=3x = 4,\ y = 3
Question 3
2 markseasy
Solve the simultaneous equations x+2y=8x + 2y = 8 and x−2y=0x - 2y = 0.

Worked solution

  1. Add the equations to eliminate y

    (x+2y)+(x−2y)=8+0⇒2x=8(x + 2y) + (x - 2y) = 8 + 0 \Rightarrow 2x = 8

    The y-terms have opposite signs, so they cancel when you add. That leaves one equation in x only.

  2. Solve for x

    x=82=4x = \frac{8}{2} = 4

    Divide both sides by 2.

  3. Substitute back to find y

    (4)+2y=8⇒2y=4⇒y=2(4) + 2y = 8 \Rightarrow 2y = 4 \Rightarrow y = 2

    Put x=4x = 4 into equation (1) and solve the one-step equation that is left. The solution is x=4x = 4, y=2y = 2.

Answer
x=4, y=2x = 4,\ y = 2
Question 4
2 markseasy
Solve the simultaneous equations 3x+y=103x + y = 10 and x+y=6x + y = 6.

Worked solution

  1. Subtract the equations to eliminate y

    (3x+y)−(x+y)=10−6⇒2x=4(3x + y) - (x + y) = 10 - 6 \Rightarrow 2x = 4

    The y-terms have the same sign, so they cancel when you subtract. That leaves one equation in x only.

  2. Solve for x

    x=42=2x = \frac{4}{2} = 2

    Divide both sides by 2.

  3. Substitute back to find y

    3(2)+y=10⇒y=43(2) + y = 10 \Rightarrow y = 4

    Put x=2x = 2 into equation (1) and solve the one-step equation that is left. The solution is x=2x = 2, y=4y = 4.

Answer
x=2, y=4x = 2,\ y = 4
Question 5
2 markseasy
Solve the simultaneous equations x+y=5x + y = 5 and 2x−y=42x - y = 4.

Worked solution

  1. Add the equations to eliminate y

    (x+y)+(2x−y)=5+4⇒3x=9(x + y) + (2x - y) = 5 + 4 \Rightarrow 3x = 9

    The y-terms have opposite signs, so they cancel when you add. That leaves one equation in x only.

  2. Solve for x

    x=93=3x = \frac{9}{3} = 3

    Divide both sides by 3.

  3. Substitute back to find y

    (3)+y=5⇒y=2(3) + y = 5 \Rightarrow y = 2

    Put x=3x = 3 into equation (1) and solve the one-step equation that is left. The solution is x=3x = 3, y=2y = 2.

Answer
x=3, y=2x = 3,\ y = 2

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