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Solving Simultaneous Equations Graphically

discia.org 10 questions · 26 marks
Name Date
  1. Either paper 2 marks

    The graphs of \(y = -x + 7\) and \(y = x + 1\) cross at the point \((3, 4)\).

    (a)

    Write down the solution of the simultaneous equations

    \(y = -x + 7\)

    \(y = x + 1\)

    (b)

    The graph of \(y = -3x + 13\) also passes through \((3, 4)\).

    Write down the solution of the simultaneous equations

    \(y = -x + 7\)

    \(y = -3x + 13\)

    Show answer
    (a) \(x = 3\), \(y = 4\)
    (b) \(x = 3\), \(y = 4\)
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  2. Either paper 3 marks

    The graphs of \(y = 2x + 9\) and \(y = -2x - 7\) are both drawn on the grid.

    xy-8-6-4-22468-10-9-8-7-6-5-4-3-2123456789100
    (a)

    State the coordinates of the point where the two lines cross.

    (b)

    Use the graph to solve the simultaneous equations \(y = 2x + 9\) and \(y = -2x - 7\)

    Show answer
    (a) \((-4, 1)\)
    (b) \(x = -4\), \(y = 1\)
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  3. Either paper 3 marks

    Omar is solving the simultaneous equations

    \(y = -x - 4\)

    \(y = 2x - 7\)

    by drawing both graphs. Omar says the graphs cross at \((1, -5)\).

    (a)

    Calculate the value of \(y\) given by \(y = -x - 4\) when \(x = 1\).

    (b)

    Calculate the value of \(y\) given by \(y = 2x - 7\) when \(x = 1\).

    (c)

    Is his answer right? Give a reason.

    Show answer
    (a) \(-5\)
    (b) \(-5\)
    (c) Yes — the point is on both graphs
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  4. Either paper 3 marks

    The graphs of \(y = -x - 3\) and \(y = 2\) are both drawn on the grid.

    xy-8-6-4-22468-10-9-8-7-6-5-4-3-2123456789100
    (a)

    State the coordinates of the point where the two graphs cross.

    (b)

    Use the graph to solve the simultaneous equations \(y = -x - 3\) and \(y = 2\)

    Show answer
    (a) \((-5, 2)\)
    (b) \(x = -5\), \(y = 2\)
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  5. Either paper 2 marks

    The graphs of \(y = -2x - 10\) and \(y = x + 8\) are both drawn on the grid. Both axes are numbered in twos.

    xy-12-10-8-6-424681012-12-10-8-6-4-2246810120

    Use the graph to solve the simultaneous equations.

    Show answer
    \(x = -6\), \(y = 2\)
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  6. Either paper 2 marks

    The graphs of \(y = x - 5\) and \(y = x + 4\) are both drawn on the grid.

    xy-8-6-4-22468-10-9-8-7-6-5-4-3-2123456789100

    Explain why the simultaneous equations \(y = x - 5\) and \(y = x + 4\) have no solution.

    Show answer
    Both lines have gradient 1, so they are parallel and never cross — there is no point on both graphs, so no pair of values fits both equations.
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  7. Either paper 3 marks

    The graph of \(y = -x + 2\) has been drawn on the grid.

    (a) xy-8-6-4-22468-10-9-8-7-6-5-4-3-2123456789100

    On the same grid, draw the graph of \(y = x - 6\).

    (b)

    Use the two graphs to solve the simultaneous equations

    \(y = -x + 2\)

    \(y = x - 6\)

    Show answer
    (a) A straight line through \((0, -6)\) and \((4, -2)\), with gradient \(1\)
    (b) \(x = 4\), \(y = -2\)
    (a)xy-8-6-4-22468-10-9-8-7-6-5-4-3-2123456789100
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  8. Either paper 2 marks

    The graphs of \(y = 3x - 9\) and \(y = -2x - 4\) cross at the point \((1, -6)\).

    (a)

    Write down the solution of the simultaneous equations

    \(y = 3x - 9\)

    \(y = -2x - 4\)

    (b)

    The graph of \(y = -3x - 3\) also passes through \((1, -6)\).

    Write down the solution of the simultaneous equations

    \(y = 3x - 9\)

    \(y = -3x - 3\)

    Show answer
    (a) \(x = 1\), \(y = -6\)
    (b) \(x = 1\), \(y = -6\)
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  9. Either paper 3 marks

    The graphs of \(y = -x - 2\) and \(y = x + 4\) are both drawn on the grid.

    xy-8-6-4-22468-10-9-8-7-6-5-4-3-2123456789100
    (a)

    State the coordinates of the point where the two lines cross.

    (b)

    Use the graph to solve the simultaneous equations \(y = -x - 2\) and \(y = x + 4\)

    Show answer
    (a) \((-3, 1)\)
    (b) \(x = -3\), \(y = 1\)
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  10. Either paper 3 marks

    Tomas is solving the simultaneous equations

    \(y = 3x - 7\)

    \(y = -x - 3\)

    by drawing both graphs. Tomas says the graphs cross at \((-1, -10)\).

    (a)

    Work out the value of \(y\) given by \(y = 3x - 7\) when \(x = -1\).

    (b)

    Work out the value of \(y\) given by \(y = -x - 3\) when \(x = -1\).

    (c)

    Is his answer right? Give a reason.

    Show answer
    (a) \(-10\)
    (b) \(-2\)
    (c) No — the point is on the first graph but not on the second
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discia.org Solving Simultaneous Equations Graphically · sheet 1
Answers Solving Simultaneous Equations Graphically · sheet 1
  1. (a) \(x = 3\), \(y = 4\)
    (b) \(x = 3\), \(y = 4\)
  2. (a) \((-4, 1)\)
    (b) \(x = -4\), \(y = 1\)
  3. (a) \(-5\)
    (b) \(-5\)
    (c) Yes — the point is on both graphs
  4. (a) \((-5, 2)\)
    (b) \(x = -5\), \(y = 2\)
  5. \(x = -6\), \(y = 2\)
  6. Both lines have gradient 1, so they are parallel and never cross — there is no point on both graphs, so no pair of values fits both equations.
  7. (a) A straight line through \((0, -6)\) and \((4, -2)\), with gradient \(1\)
    (b) \(x = 4\), \(y = -2\)
    (a)xy-8-6-4-22468-10-9-8-7-6-5-4-3-2123456789100
  8. (a) \(x = 1\), \(y = -6\)
    (b) \(x = 1\), \(y = -6\)
  9. (a) \((-3, 1)\)
    (b) \(x = -3\), \(y = 1\)
  10. (a) \(-10\)
    (b) \(-2\)
    (c) No — the point is on the first graph but not on the second
discia.org Solving Simultaneous Equations Graphically · sheet 1