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Cambridge IGCSE0607

Equations of linear graphs

International Mathematics 0607 Chapter Notes

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Equations of linear graphs
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In this chapter

  1. 1The line y = mx + c
  2. 2Vertical and horizontal lines, x = k
  3. 3Finding the equation of a line
  4. 4Rearranging to find the gradient and y-intercept
  5. 5Equations in the form ax + by = c

1. The line y = mx + c

A straight-line graph has an equation of the form y = mx + c. The number m is the gradient and c is the y-intercept, where the line crosses the y-axis.

To read them off, make sure y is on its own. In y = 6x + 3 the gradient is 6 and the y-intercept is 3. A positive m means the line slopes up. A negative m means it slopes down.

y = mx + c

Examiner insight

“Many did not know how to find the gradient of the line from its equation. A small number gave 2x as the gradient.” “Diagram B (which could be, for example, x = 3) was most frequently chosen instead of A for y = 2x.”

Worked example 13 marks

Line L has equation y = 2x - 5.

(a) Line L crosses the y-axis at point A. Write down the coordinates of A.

(b) Write down the gradient of line L.

(c) Point B has coordinates (4, 3). Show that line L passes through B.

  1. 1

    (a) the y-intercept is -5, so A = (0, -5) [1]

  2. 2

    (b) gradient = 2 [1]

  3. 3

    (c) substituting x = 4: y = 2 × 4 - 5 = 3, so the line passes through (4, 3) [1]

2. Vertical and horizontal lines, x = k

A line with equation x = k is parallel to the y-axis. Every point on it has the same x-coordinate k. It crosses the x-axis at (k, 0).

A line with equation y = c is horizontal. It is y = mx + c with m = 0.

x = k

y = c

Examiner insight

“Others correctly said that the given line is horizontal when x = 5 would be vertical.” “Some incorrectly said the given line is on the y-axis when it goes through the y-axis.”

Worked example 12 marks

The point P has coordinates (3, 5). Write down the equation of

(a) the vertical line through P,

(b) the horizontal line through P.

  1. 1

    (a) every point on a vertical line has x-coordinate 3: x = 3 [1]

  2. 2

    (b) every point on a horizontal line has y-coordinate 5: y = 5 [1]

3. Finding the equation of a line

To find the equation from a graph, first read the y-intercept c where the line crosses the y-axis. Then find the gradient from two points on the line.

Write the answer as y = mx + c in fully simplified form.

m = (y₂ - y₁)/(x₂ - x₁)

y = mx + c

Examiner insight

“The simplest mark to obtain was for the realisation that, in the y = mx + c form of the line, the value of c is the y-coordinate when x = 0, given by the point (0, –3) in the question.” “Only a few candidates wrote the formula the wrong way around with the xs on top.”

Worked example 13 marks

Do not use a calculator. A straight line passes through (0, 4) and (3, -2). Find its equation in the form y = mx + c.

  1. 1

    gradient m = (-2 - 4)/(3 - 0) = (-6)/3 = -2 [1]

  2. 2

    the line crosses the y-axis at (0, 4), so c = 4 [1]

  3. 3

    y = -2x + 4 [1]

4. Rearranging to find the gradient and y-intercept

An equation may be given as ax + by = c. To read the gradient and y-intercept, rearrange it into y = mx + c first.

Do the same operation to every term. Dividing by a negative number changes the sign of every term.

ax + by = c

Examiner insight

“Candidates either did not attempt to rearrange the equation as shown by answers of –6 or had difficulty rearranging the equation giving –3 as their answer.”

Worked example 13 marks

The equation of a line is 4y - x = 12. Find the gradient and the y-intercept of the line.

  1. 1

    4y = x + 12, so y = 1/4x + 3 [1]

  2. 2

    gradient = 1/4 [1]

  3. 3

    y-intercept = 3 [1]

5. Equations in the form ax + by = c

Some questions ask for the equation in a different form, such as ax + by = c with whole numbers a, b and c.

Find y = mx + c first. If m or c is a fraction, multiply every term by the common denominator, then collect the x and y terms on one side.

Examiner insight

“It was then very common for candidates to struggle to rearrange this equation into the correct form where a, b and k were integers.”

Worked example 14 marks

Do not use a calculator. A line passes through (0, 2) and (4, -1). Find its equation in the form ax + by = c, where a, b and c are integers.

  1. 1

    gradient = (-1 - 2)/(4 - 0) = -3/4 [1]

  2. 2

    y-intercept is 2, so y = -3/4x + 2 [1]

  3. 3

    multiply every term by 4: 4y = -3x + 8 [1]

  4. 4

    3x + 4y = 8 [1]

Recap

  • y = mx + c: m is the gradient and c is the y-intercept.
  • x = k is a vertical line, parallel to the y-axis. y = c is horizontal.
  • Gradient = (y₂ - y₁)/(x₂ - x₁), with the y-differences on top.
  • Give the answer as y = mx + c in fully simplified form.
  • Extended: rearrange ax + by = c into y = mx + c before reading m and c.
  • Extended: for whole-number a, b, c, multiply through to clear fractions.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. Write down the equation of the vertical line that passes through (-4, 7).1 mark
  2. Do not use a calculator. Write down the gradient of the line y = 3x - 7 and the coordinates of the point where it crosses the y-axis.2 marks
  3. Without drawing the graph, state whether the line y = 4 - 3x slopes up or down as x increases. Give its gradient and y-intercept.2 marks
  4. Do not use a calculator. Show that the point (3, 11) lies on the line y = 4x - 1. Does the point (2, 8) lie on it?3 marks
  5. Do not use a calculator. A line passes through (0, -2) and (4, 6). Find its equation in the form y = mx + c.3 marks
  6. A line passes through (0, 5) and (2, 1). Find its equation in the form y = mx + c.3 marks
  7. Do not use a calculator. Find the gradient and the y-intercept of the line 2x + y = 9.2 marks
  8. Do not use a calculator. Find the gradient and the y-intercept of the line 6x - 3y = 12.3 marks
  9. Do not use a calculator. On a map grid, a straight road has equation 5x - 2y = 20 and a straight fence has equation x = 6. (a) Find the gradient and the y-intercept of the road. (b) Find the coordinates of the point where the road meets the fence. (c) A gate is at (2, -5). Show that the gate is on the road.6 marks
  10. On a site plan, a straight cable passes through P(-3, 4) and Q(6, -2). (a) Find the equation of the cable in the form ax + by = c, where a, b and c are integers. (b) A wall has equation x = 9. Find the coordinates of the point where the cable meets the wall. (c) Find the coordinates of the point where the cable crosses the x-axis.7 marks

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