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Cambridge IGCSE0607

Parallel lines

International Mathematics 0607 Chapter Notes

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Parallel lines
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In this chapter

  1. 1Parallel lines have the same gradient
  2. 2Equation of the parallel line through a point
  3. 3Lines not given as y = mx + c
  4. 4Parallel to a line through two points

1. Parallel lines have the same gradient

A straight line with equation y = mx + c has gradient m, the coefficient of x. The line crosses the y-axis at c.

Parallel lines have the same gradient. The constant terms have no bearing on the gradient, so choosing a different constant gives a parallel line.

y = mx + c

parallel lines: m₁ = m₂

Examiner insight

“The coefficient of the x-term is the gradient, so the two lines should have the same x-coefficient and the constant terms have no bearing on the gradient.” “An answer of y = 3x + c where c is not defined, does not get the mark as candidates must choose a value for c which can be anything except +5.”

Worked example 12 marks

Write down the equation of the line parallel to y = 5x + 2 that crosses the y-axis at (0, -4).

  1. 1

    gradient of the given line = 5, so m = 5 [1]

  2. 2

    c = -4, so y = 5x - 4 [1]

2. Equation of the parallel line through a point

The parallel line has the same gradient m as the given line. Put the coordinates of the point into y = mx + c and solve for c.

Write the equation y = mx + c with a value in place of both m and c.

c = y - mx

Examiner insight

“Here, the gradient is equal to that of the parallel line so only the place where the line L crosses the y-axis needs to be found.” “Candidates may find it helpful to draw a diagram to aid them in visualising the situation; very few diagrams were seen.”

Worked example 13 marks

Do not use a calculator.

Find the equation of the line parallel to y = 4x - 1 that passes through (1, -3).

  1. 1

    parallel, so m = 4 [1]

  2. 2

    -3 = 4 × 1 + c, so c = -7 [1]

  3. 3

    y = 4x - 7 [1]

3. Lines not given as y = mx + c

If the equation is not in the form y = mx + c, rearrange it first. Get y on its own.

Whatever you do to one term, do to every term. The gradient is then the coefficient of x.

3y = 6x + 9 ⇒ y = 2x + 3

Examiner insight

“The most common error was first to divide through by 3 but most candidates forgot to divide the x as well as the numbers.” “Those who rearranged the equation by moving the x first usually managed to reach the correct answer.”

Worked example 14 marks

Do not use a calculator.

Find the equation of the line parallel to 2x + y = 9 that passes through (1, 5).

  1. 1

    rearrange: y = -2x + 9 [1]

  2. 2

    gradient m = -2 [1]

  3. 3

    5 = -2 × 1 + c, so c = 7 [1]

  4. 4

    y = -2x + 7 [1]

4. Parallel to a line through two points

When the given line passes through two points, find its gradient from them. The parallel line has the same gradient.

Put the change in y on top and the change in x underneath.

m = (y₂ - y₁)/(x₂ - x₁)

Examiner insight

“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.

The line AB passes through A(1, 2) and B(4, 11). Find the equation of the line parallel to AB that passes through (0, -5).

  1. 1

    m = (11 - 2)/(4 - 1) = 3 [1]

  2. 2

    the line crosses the y-axis at (0, -5), so c = -5 [1]

  3. 3

    y = 3x - 5 [1]

Recap

  • In y = mx + c, m is the gradient and the line crosses the y-axis at c.
  • Parallel lines have the same gradient.
  • To find the parallel line through a point: take m, put the point into y = mx + c, solve for c.
  • Rearrange to y = mx + c first, dividing every term.
  • Gradient from two points: change in y over change in x.

End-of-chapter exercise

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

  1. Write down the gradient of a line parallel to y = 6x - 5.1 mark
  2. Are the lines y = 2x + 3 and y = 3x + 2 parallel? Give a reason.2 marks
  3. Write down the equation of the line parallel to y = -3x + 4 that crosses the y-axis at (0, 1).2 marks
  4. Do not use a calculator. Find the equation of the line parallel to y = 2x + 5 that passes through (3, 4).3 marks
  5. Do not use a calculator. Find the equation of the line parallel to y = 1/2x - 3 that passes through (4, 1).3 marks
  6. Find the equation of the line parallel to 4x + y = 10 that passes through the origin.2 marks
  7. Find the gradient of a line parallel to 4y = 8x - 12.2 marks
  8. Do not use a calculator. Find the equation of the line parallel to 2x + 3y = 6 that passes through (3, -4).4 marks
  9. Do not use a calculator. On a map grid, a straight road lies along the line 3y - 6x = 12. A fence lies along the line ky = 8x - 5. The fence is parallel to the road. (a) Find the value of k. (b) A cycle path is parallel to the road and passes through the point (4, -1). Find the equation of the cycle path in the form y = mx + c.5 marks
  10. Do not use a calculator. ABCD is a parallelogram. AB is parallel to DC and AD is parallel to BC. A is the point (-2, 1), B is the point (4, 4) and D is the point (2, 9). (a) Find the equation of the line through D and C. (b) Find the equation of the line through B and C. (c) Show that the point (8, 12) lies on both of these lines.7 marks

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