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Co-Ordinate Geometry

Co-Ordinate Geometry

The equation of a straight line

Equation Of A Line Core + Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Reading y = mx + c
  4. Finding the equation of a line
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 3.2, 3.5: the equation of a straight line
  • 3.5Use \(y=mx+c\): m is the gradient, c is the y-intercept; rearrange \(ax+by=c\) into this form.
  • 3.5Find the equation of a line from a gradient and a point, from two points, or from a graph.
  • 3.2Find where a line crosses the axes and test whether a point lies on it.

Before you start: Gradient and rearranging. Next: Parallel and perpendicular lines.

Formula bank

Straight lineLearn it\(y=mx+c\)m = gradient, c = where it crosses the y-axis.Use it when any non-vertical line.
Special linesLearn it\(x=k\) (vertical), \(y=k\) (horizontal, gradient 0)Use it when lines parallel to an axis.
Point on a line?Learn itsubstitute its x; does the equation give its y?Use it when 'does (6, −5) lie on the line?'.

Reading y = mx + c

Start at the y-intercept \((0,\ c)\). Each step of 1 to the right changes y by \(m\). So the line is \(y=mx+c\). An equation like \(2x+3y=12\) must be rearranged first: \(y=-\tfrac23x+4\), gradient \(-\tfrac23\), y-intercept 4.

two lines-2246-4-22468xyy = 3x − 22x + 3y = 12
Blue: m = 3, c = −2. Pink: m = −2/3, c = 4.

In \(2x+3y=12\) the gradient is NOT 2: make \(y\) the subject first.

Try it yourself

1
Gradient of \(4x+2y=9\)?
Hint
\(2y=-4x+9\).
Show answer

Answer −2

2
y-intercept of \(y=3x-2\)?
Show answer

Answer −2

Finding the equation of a line

General methodHow to find the equation of a line

  1. Find the gradient m (given, or from two points: Δy ÷ Δx).
  2. Put m and one point \((x,\ y)\) into \(y=mx+c\) and solve for c.
  3. Write \(y=mx+c\) with the numbers.
  4. Check with the other point.

Example Find the equation of the line through (1, −2) and (4, 7).

Gradient first, then c.

  1. \(m=\dfrac{7-(-2)}{4-1}=3\)
  2. \(-2=3(1)+c\Rightarrow c=-5\)

Answer \(y=3x-5\)

Try it yourself

1
Line through (0, 1) and (2, 7)?
Show answer

Answer \(y=3x+1\)

2
Line with gradient 3 through (2, 5)?
Show answer

Answer \(y=3x-1\)

Ways they test it

Type 1 Gradient and y-intercept

"Write down the gradient of \(5x+2y=15\)." (1 mark)

Spot it an equation not in y = mx + c form.

  1. Make y the subject.
  2. Read m.

For example \(2y=-5x+15\)

  1. \(y=-2.5x+7.5\)

Answer −2.5

Type 2 Line through two points

"Find the equation of the line through (−1, 6) and (3, −2)." (3 marks)

Spot it two points.

  1. Gradient.
  2. c from one point.
  3. y = mx + c.

For example \(m=\dfrac{-8}{4}=-2\)

  1. \(6=-2(-1)+c\Rightarrow c=4\)

Answer \(y=-2x+4\)

Type 3 Point on the line?

"Does (5, −6) lie on y = −2x + 4?" (1 mark)

Spot it a point to test.

  1. Substitute x; compare y.

For example \(-2(5)+4\)

    Answer −6: yes

    Type 4 Intercepts

    "Where does \(3x-4y=12\) cross the axes?"

    Spot it 'crosses the axes'.

    1. x = 0 gives the y-intercept; y = 0 gives the x-intercept.

    For example \(3x-4y=12\)

      Answer (0, −3) and (4, 0)

      Mistakes to avoid

      • ✗ gradient of \(2x+3y=12\) is 2✓ \(-\tfrac23\)Rearrange to y = … first.
      • ✗ c = the x-intercept✓ c = the y-interceptc is where x = 0.
      • ✗ \(-2=3(1)+c\Rightarrow c=1\)✓ \(c=-5\)Subtract 3 from −2 carefully.

      Summary

      • \(y=mx+c\): gradient m, y-intercept c.
      • Rearrange ax + by = c to y = … to read m and c.
      • Equation: gradient, then c from a point.
      • Test a point by substituting.

      If you see …

      • "find the equation of the line"gradient first, then c.
      • x and y on the same sidemake y the subject.

      Practice

      Practice: easy, medium, hard

      Easy

      1
      Gradient of \(y=-\tfrac32x+4\)?
      Show answer

      Answer −1.5

      2
      Gradient of \(5x+2y=15\)?
      Show answer

      Answer −2.5

      3
      Where does \(3x-4y=12\) cross the y-axis? Type the y-value.
      Show answer

      Answer −3

      Medium

      1
      Find the equation of the line through (−1, 6) and (3, −2).
      Guide me

      What is it asking? an equation

      What do we know? \(m=\tfrac{-8}{4}\)

      Plan c from (−1, 6)

      Show working
      1. \(m=-2\)
      2. \(6=2+c\Rightarrow c=4\)

      Answer \(y=-2x+4\)

      2
      Find the equation of the line through (0, 3) and (4, 0).
      Guide me

      What is it asking? an equation

      What do we know? c = 3 (it passes through (0, 3))

      Plan \(m=-\tfrac34\)

      Show working
      1. \(m=\dfrac{0-3}{4-0}=-\tfrac34\)
      2. \(y=-\tfrac34x+3\)

      Answer \(y=-\tfrac34x+3\)

      3
      Does (5, −6) lie on y = −2x + 4? Type yes or no.
      Guide me

      What is it asking? yes or no

      What do we know? substitute x = 5

      Plan compare with −6

      Show working
      1. \(-10+4=-6\) ✓

      Answer yes

      Hard

      1
      Find the equation of the line through (−2, 7) and (4, −2).
      Guide me

      What is it asking? an equation

      What do we know? \(m=\tfrac{-9}{6}=-\tfrac32\)

      Plan c from (−2, 7)

      Show working
      1. \(7=-\tfrac32(-2)+c=3+c\)
      2. \(c=4\)

      Answer \(y=-1.5x+4\)

      PartAnswer / workingMarksGuidance
      \(m=-\tfrac32\)B1
      \(c=4\)M1
      \(y=-1.5x+4\)A1
      Total: 3 marks
      2
      The line in the previous question: does (6, −5) lie on it? Type yes or no.
      Guide me

      What is it asking? yes or no

      What do we know? substitute x = 6

      Plan \(-9+4\)

      Show working
      1. \(-1.5(6)+4=-5\) ✓

      Answer yes

      PartAnswer / workingMarksGuidance
      yes, with workingB1
      Total: 1 mark
      3
      Find where \(3x-4y=12\) crosses the x-axis. Type the x-value.
      Guide me

      What is it asking? an x-value

      What do we know? on the x-axis, y = 0

      Plan \(3x=12\)

      Show working
      1. \(3x=12\Rightarrow x=4\)

      Answer 4

      PartAnswer / workingMarksGuidance
      4B1
      Total: 1 mark

      Can I do this?

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

      Quiz

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