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Equations of Straight Lines

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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 straight line L passes through the points (8,\,-1) and (-4,\,8)

Work out the equation of L.

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

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

Hint

Find the gradient first (change in y divided by change in x), then use y-y_1=m(x-x_1) and clear the fraction.

Worked solution
  1. Gradient =\dfrac{8-(-1)}{-4-8}=\dfrac{9}{-12}=-\dfrac34
  2. Use y-y_1=m(x-x_1) with (8,\,-1): y+1=-\dfrac34(x-8)
  3. Multiply both sides by 4: 4y+4=-3(x-8)=-3x+24
  4. Collect x and y on the left: 3x+4y=20
  5. Check with (-4,\,8): -12+32=20 ✓
  6. Answer: 3x+4y=20

Question 22 marks

The line with equation ky-10x=3 is parallel to the line y=4x-1

k is a constant.

Work out the value of k.

Hint

Rearrange ky-10x=3 into the form y=mx+c; parallel lines have the same gradient.

Worked solution
  1. Rearrange: ky=10x+3, so y=\dfrac{10}{k}x+\dfrac{3}{k}
  2. The gradient is \dfrac{10}{k}
  3. Parallel lines have equal gradients: \dfrac{10}{k}=4
  4. k=\dfrac{10}{4}=\dfrac52
  5. Answer: k=\dfrac52

Question 3Challenge6 marks

P is the point (-4,\,3) and Q is the point (6,\,-2)

R is the point on PQ such that PR:RQ=4:1

The line L passes through R and is perpendicular to PQ.

(a)

Work out the coordinates of R.

2 marks

Write your answer as (x, y)

(b)

Work out the equation of L.

Give your answer in the form y=mx+c

2 marks

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

(c)

The line L meets the line x+3y=8 at the point S.

Work out the coordinates of S.

2 marks

Write your answer as (x, y)

Hint

R is \dfrac45 of the way from P to Q; the gradient of L is the negative reciprocal of the gradient of PQ.

Worked solution

Part (a)

  1. From P to Q: x goes up by 6-(-4)=10 and y changes by -2-3=-5
  2. PR:RQ=4:1, so R is \dfrac45 of the way from P to Q
  3. \dfrac45\times10=8 and \dfrac45\times(-5)=-4
  4. R=(-4+8,\ 3-4)
  5. Answer: (4,\,-1)

Part (b)

  1. Gradient of PQ=\dfrac{-2-3}{6-(-4)}=\dfrac{-5}{10}=-\dfrac12
  2. Perpendicular gradient =2 (because -\dfrac12\times2=-1)
  3. Through R(4,\,-1): y+1=2(x-4)
  4. y=2x-8-1
  5. Answer: y=2x-9

Part (c)

  1. Substitute y=2x-9 into x+3y=8: x+3(2x-9)=8
  2. x+6x-27=8
  3. 7x=35, so x=5
  4. y=2\times5-9=1
  5. Answer: (5,\,1)

More on this topic: Equations of Straight Lines worksheet with full solutions

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