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Graphical Solutions

Graphical Solutions

Solving equations with graphs

Case 1 Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Where graphs meet
  4. Which line should I draw?
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 2.10: solving equations with graphs
  • 2.10Solve \(f(x)=0\) and \(f(x)=k\) from a graph.
  • 2.10Solve simultaneous equations from where two graphs intersect.
  • 2.10Find the straight line to draw on a curve to solve a NEW equation, and use it.

Before you start: Drawing and reading graphs. Next: Gradients and tangents.

Formula bank

IntersectionsLearn itwhere \(y=f(x)\) meets \(y=g(x)\): \(f(x)=g(x)\)the x-coordinates of the crossing points are the solutions.Use it when any 'use your graph to solve'.
The line to drawLearn itline \(=\) curve \(-\) (new equation's left side, when it equals 0)e.g. curve \(x^{2}-2x-3\), new \(x^{2}-3x-1=0\): line \(y=x-2\).Use it when 'by drawing a suitable straight line'.

Where graphs meet

A point on both \(y=f(x)\) and \(y=g(x)\) has the same y on both, so \(f(x)=g(x)\) there. The x-coordinates of the intersections solve that equation.

To solveDrawRead
\(f(x)=0\)nothing extra (the x-axis)where the curve crosses the x-axis
\(f(x)=k\)the horizontal line \(y=k\)x where they meet
\(f(x)=mx+c\)the line \(y=mx+c\)x where they meet
two equations in x and yboth linesthe crossing point (x and y)
y = x² − 2x − 3 with lines-3-2-112345-5-4-3-2-112345678xy
y = x² − 2x − 3 with y = 2 (pink) and y = x − 2 (green).
  1. x-axis: \(x=-1\), \(3\) solve \(x^{2}-2x-3=0\).
  2. \(y=2\): \(x\approx-1.45\), \(3.45\) solve \(x^{2}-2x-3=2\).
  3. \(y=x-2\): \(x\approx-0.3\), \(3.3\) solve \(x^{2}-3x-1=0\).

Which line should I draw?

General methodHow to find the line to draw

  1. Write the curve you have: \(y=\) (curve).
  2. Write the new equation as (expression) \(=0\).
  3. Subtract: \(y=\) (curve) \(-\) (expression). The \(x^{2}\) (or \(x^{3}\)) terms cancel, leaving a straight line.
  4. Draw that line on the grid (two or three points).
  5. Read the x-coordinate of every crossing.

Example The curve \(y=x^{2}-2x-3\) is drawn. Which line solves \(x^{2}-4x+1=0\)?

Curve minus the new expression.

  1. \((x^{2}-2x-3)-(x^{2}-4x+1)\)
  2. \(=2x-4\)

Answer \(y=2x-4\)

Try it yourself

1
Curve \(y=x^{2}-2x-3\). Line to draw to solve \(x^{2}-3x-1=0\)?
Show answer

Answer \(y=x-2\)

2
Curve \(y=x^{2}+x-2\). Line to draw to solve \(x^{2}-2x-5=0\)?
Show answer

Answer \(y=3x+3\)

3
Curve \(y=x^{3}-3x+1\). Line to draw to solve \(x^{3}-4x+2=0\)?
Show answer

Answer \(y=x-1\)

Ways they test it

Type 1 Solve f(x) = k

"Use your graph to solve \(x^{2}-2x-3=2\)." (2 marks)

Spot it a number on the right.

  1. Draw y = k.
  2. Read both x-values.

Type 2 By drawing a suitable line

"By drawing a suitable straight line, solve \(x^{2}-4x+1=0\)." (3 marks)

Spot it 'suitable straight line'.

  1. Curve − new expression = the line.
  2. Draw it.
  3. Read every crossing.

For example line \(y=2x-4\)

    Answer \(x\approx0.27,\ 3.73\)

    Type 3 Find the line only

    "Find the equation of the line you would draw ..." (1-2 marks)

    Spot it no drawing asked.

    1. Subtract and simplify.

    Type 4 Simultaneous by graph

    "Draw \(x+y=8\) and \(y=2x-1\). Use them to solve the simultaneous equations." (3 marks, Core)

    Spot it two straight lines.

    1. Two points per line.
    2. Read x and y at the crossing.

    For example lines cross at (3, 5)

      Answer \(x=3,\ y=5\)

      Mistakes to avoid

      • ✗ Subtracting the wrong way round✓ curve − new expressionOtherwise every sign flips.
      • ✗ Giving y-values✓ give the x-valuesThe equation is in x.
      • ✗ Reading only one crossing✓ all of themA line can meet a parabola twice.

      Summary

      • Intersections solve 'curve = line'.
      • f(x) = 0: x-axis crossings; f(x) = k: the line y = k.
      • Line to draw = curve − (expression that equals 0).
      • Read every intersection, to the nearest small square.

      If you see …

      • "by drawing a suitable straight line"subtract to find the line, draw it, read the crossings.
      • two linear equationsdraw both; the crossing gives x and y.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      \(y=x^{2}-2x-3\) is drawn. Solve \(x^{2}-2x-3=0\).
      Show answer

      Answer −1, 3

      2
      \(y=x^{2}-2x-3\) is drawn. Which line solves \(x^{2}-2x-3=2\)? Type y = ….
      Show answer

      Answer \(y=2\)

      3
      Solve \(x+y=8\) and \(y=2x-1\) (where the lines cross).
      Show answer

      Answer \(x=3,\ y=5\)

      Medium

      1
      Curve \(y=x^{2}-2x-3\). Find the line to solve \(x^{2}-4x+1=0\).
      Guide me

      What is it asking? a line

      What do we know? curve − new expression

      Plan \((x^{2}-2x-3)-(x^{2}-4x+1)\)

      Show working
      1. \(2x-4\)

      Answer \(y=2x-4\)

      2
      Using that line, solve \(x^{2}-4x+1=0\) to 2 d.p.
      Guide me

      What is it asking? two x-values

      What do we know? where \(y=2x-4\) meets the curve

      Plan read both crossings

      Show working
      1. ≈ 0.27 and 3.73

      Answer 0.27, 3.73

      3
      \(y=x^{2}-2x-3\) is drawn. Solve \(x^{2}-2x-3=2\) to 2 d.p.
      Guide me

      What is it asking? two x-values

      What do we know? draw y = 2

      Plan read the crossings

      Show working
      1. ≈ −1.45 and 3.45

      Answer −1.45, 3.45

      Hard

      1
      Curve \(y=x^{2}+x-2\). Find the line to draw to solve \(x^{2}-2x-5=0\).
      Guide me

      What is it asking? a line

      What do we know? subtract

      Plan \((x^{2}+x-2)-(x^{2}-2x-5)\)

      Show working
      1. \(3x+3\)

      Answer \(y=3x+3\)

      PartAnswer / workingMarksGuidance
      \(y=3x+3\)B2
      Total: 2 marks
      2
      Curve \(y=x^{3}-3x+1\). Find the line to draw to solve \(x^{3}-4x+2=0\).
      Guide me

      What is it asking? a line

      What do we know? subtract

      Plan \((x^{3}-3x+1)-(x^{3}-4x+2)\)

      Show working
      1. \(x-1\)

      Answer \(y=x-1\)

      PartAnswer / workingMarksGuidance
      \(y=x-1\)B2
      Total: 2 marks
      3
      \(y=x^{2}-2x-3\) is drawn. By drawing a suitable line, solve \(x^{2}-3x-1=0\) (1 d.p.).
      Guide me

      What is it asking? two x-values

      What do we know? line \(y=x-2\)

      Plan read the crossings

      Show working
      1. line \(y=x-2\)
      2. ≈ −0.3 and 3.3

      Answer −0.3, 3.3

      PartAnswer / workingMarksGuidance
      \(y=x-2\) drawnB1
      −0.3B1
      3.3B1
      Total: 3 marks

      Can I do this?

      Anything unticked: revise that section first. All ticked: take the quiz.

      Quiz

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