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Simultaneous Equations (Two Unknowns)

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AQA Level 2 Further Maths practice questions, each with a hint and a full worked solution.

Have a pen and paper handy, to work out your answers.

Question 13 marks

The lines 5x+2y=3 and 3x-4y=20 meet at the point P.

Work out the coordinates of P.

Write your answer as (x, y)

Hint

Multiply the first equation by 2 so the y terms can be eliminated by adding.

Worked solution
  1. The point where two lines meet satisfies both equations, so solve them simultaneously.
  2. Double the first equation: 10x+4y=6
  3. Add to 3x-4y=20: 13x=26, so x=2
  4. Substitute into 5x+2y=3: 10+2y=3, so 2y=-7 and y=-3.5
  5. Check in the second equation: 3(2)-4(-3.5)=6+14=20 ✓
  6. Answer: P=(2,\,-3.5)

Question 21 mark

The line y=x+1 meets the circle x^2+y^2=61

Substituting y=x+1 into x^2+y^2=61 gives an equation that the x-coordinates of the intersection points satisfy.

Select that equation.

Hint

Write (x+1)^2 as (x+1)(x+1) and expand all four terms before simplifying.

Worked solution
  1. x^2+(x+1)^2=61
  2. (x+1)^2=(x+1)(x+1)=x^2+2x+1 (not x^2+1: you can't square term by term)
  3. x^2+x^2+2x+1=61
  4. 2x^2+2x-60=0
  5. Divide by 2: x^2+x-30=0
  6. Answer: x^2+x-30=0

Question 3Challenge6 marks

The line L passes through the point (5,\,5) and has gradient -\dfrac{1}{2}

L meets the circle x^2+y^2=170 at the points A and B.

A has a positive x-coordinate and B has a negative x-coordinate.

(a)

Work out the equation of L.

Give your answer in the form ax+by=c, where a, b and c are integers.

2 marks

Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

(b)

Work out the coordinates of A.

3 marks

Write your answer as (x, y)

(c)

Write down the coordinates of B.

1 mark

Write your answer as (x, y)

Hint

Find the equation of L first, then make x the subject (to avoid fractions) and substitute into the circle equation.

Worked solution

Part (a)

  1. Use y-y_1=m(x-x_1): y-5=-\dfrac{1}{2}(x-5)
  2. Multiply by 2: 2y-10=-(x-5)=-x+5
  3. Rearrange: x+2y=15
  4. Answer: x+2y=15

Part (b)

  1. From L: x=15-2y
  2. Substitute into the circle: (15-2y)^2+y^2=170
  3. (15-2y)(15-2y)=225-60y+4y^2, so 225-60y+4y^2+y^2=170
  4. 5y^2-60y+55=0, and dividing by 5: y^2-12y+11=0
  5. (y-1)(y-11)=0, so y=1 or y=11
  6. y=1 gives x=15-2=13; y=11 gives x=15-22=-7
  7. A has the positive x-coordinate.
  8. Answer: A=(13,\,1)

Part (c)

  1. From part (b), the other solution is y=11, x=-7.
  2. Check: (-7)^2+11^2=49+121=170 ✓ and -7+2(11)=15 ✓
  3. Answer: B=(-7,\,11)

More on this topic: Simultaneous Equations (Two Unknowns) worksheet with full solutions

All AQA Level 2 Further Maths practice questions

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