[discia] GCSE Maths › Algebra › Gradient of a Line

Gradient of a Line

discia.org 10 questions · 30 marks
Name Date
  1. Either paper 2 marks

    The grid shows the points \(C(-5, 1)\) and \(D(-3, 3)\).

    -6-5-4-3-2123456-7-6-5-4-3-212345670xyCD

    Find the gradient of the line \(CD\).

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    \(1\)
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  2. Either paper 2 marks

    The line through \(A(-3, 2)\) and \(B(-2, -1)\) is drawn on the grid.

    -6-5-4-3-2123456-7-6-5-4-3-212345670xyAB

    Calculate the gradient of the line.

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    \(-3\)
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  3. Either paper 2 marks

    A straight line has equation \(y = -x - 7\)

    (a)

    Write down the gradient of the line.

    (b)

    Write down the coordinates of the point where the line crosses the \(y\)-axis.

    Show answer
    (a) \(-1\)
    (b) \((0, -7)\)
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  4. Either paper 4 marks

    A taxi firm charges a fixed amount for every journey, plus an extra amount for each mile travelled.

    The graph shows the cost of a journey of up to \(14\) miles.

    2468101214510152025300distance (miles)cost (£)
    (a)

    State the amount the taxi firm charges before it has travelled any distance at all.

    (b)

    Work out the gradient of the line.

    (c)

    Write down what the gradient tells you about the cost of a journey.

    Show answer
    (a) \(\text{£}2\)
    (b) \(2\)
    (c) The taxi firm charges £2 for each extra mile travelled.
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  5. Either paper 3 marks

    A straight line passes through \((-2, 5)\) and \((2, k)\).

    The gradient of the line is \(-3\).

    Find the value of \(k\).

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    \(-7\)
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  6. Either paper 4 marks

    A cafe makes a drink from orange squash and water.

    The number of cartons of water \((y)\) : the number of cartons of squash \((x)\) = \(3 : 2\)

    (a)

    Use this information to draw a graph on the grid to show the relationship between the number of cartons of squash and the number of cartons of water.

    246810121416182012345678910111213141516171819200cartons of squash (x)cartons of water (y)
    (b)

    Find the gradient of the line drawn in part (a).

    (c)

    Explain what this gradient represents.

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    (a) A straight line through \((0, 0)\), \((2, 3)\) and \((4, 6)\)
    (b) \(\frac{3}{2}\)
    (c) The number of cartons of water needed for 1 carton of squash
    (a)246810121416182012345678910111213141516171819200cartons of squash (x)cartons of water (y)
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  7. Either paper 3 marks

    The line \(L\) has equation \(y = -x - 1\)

    Work out the equation of the line that is parallel to \(L\) and passes through the point \((4, 6)\).

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    \(y = -x + 10\)
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  8. Either paper 3 marks

    The points \(R\) and \(S\) are marked on the grid.

    -7-6-5-4-3-21234567-7-6-5-4-3-212345670xyRS

    Work out the gradient of the line \(RS\).

    Give your answer as a fraction in its simplest form.

    Show answer
    \(-\frac{3}{4}\)
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  9. Either paper 4 marks

    \(A\) is the point \((5, -9)\).

    \(B\) is the point \((8, -15)\).

    (a)

    Show that the gradient of the straight line passing through \(A\) and \(B\) is \(-2\).

    (b)

    \(C\) is the point \((264, -528)\).

    Does \(C\) lie on the straight line passing through \(A\) and \(B\)?

    You must show your working.

    Show answer
    (a) \(\frac{ -15 - (-9) }{ 8 - 5 } = \frac{ -6 }{ 3 } = -2\)
    (b) No: the line gives \(y = -527\) at \(x = 264\), and \(C\) has \(y = -528\)
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  10. Either paper 3 marks

    The graph gives information about the volume, \(v\) litres, of diesel in the tank of Anya's car after it has travelled a distance of \(d\) kilometres.

    501001502002503005101520253035404550550distance travelled, d (km)volume of diesel, v (litres)
    (a)

    Calculate the gradient of the graph.

    (b)

    What does the gradient of the graph represent?

    Show answer
    (a) \(-0.075\) litres per km
    (b) The diesel used per kilometre travelled (the rate of decrease of the volume): the tank loses 0.075 litres of diesel for every 1 km driven.
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discia.org Gradient of a Line · sheet 1
Answers Gradient of a Line · sheet 1
  1. \(1\)
  2. \(-3\)
  3. (a) \(-1\)
    (b) \((0, -7)\)
  4. (a) \(\text{£}2\)
    (b) \(2\)
    (c) The taxi firm charges £2 for each extra mile travelled.
  5. \(-7\)
  6. (a) A straight line through \((0, 0)\), \((2, 3)\) and \((4, 6)\)
    (b) \(\frac{3}{2}\)
    (c) The number of cartons of water needed for 1 carton of squash
    (a)246810121416182012345678910111213141516171819200cartons of squash (x)cartons of water (y)
  7. \(y = -x + 10\)
  8. \(-\frac{3}{4}\)
  9. (a) \(\frac{ -15 - (-9) }{ 8 - 5 } = \frac{ -6 }{ 3 } = -2\)
    (b) No: the line gives \(y = -527\) at \(x = 264\), and \(C\) has \(y = -528\)
  10. (a) \(-0.075\) litres per km
    (b) The diesel used per kilometre travelled (the rate of decrease of the volume): the tank loses 0.075 litres of diesel for every 1 km driven.
discia.org Gradient of a Line · sheet 1