Parallel and perpendicular lines Worked Solutions — GCSE Maths

Fully worked, step-by-step solutions to GCSE Parallel and perpendicular lines questions. See exactly how to solve problems on parallel lines, equal gradients, y = mx + c, identifying gradients.

parallel linesequal gradientsy = mx + cidentifying gradientsnegative gradientperpendicular lines
GCSE Foundation70 questionsStep-by-step solutions
Question 1
1 markeasy
A straight line is parallel to the line with equation y=3x+4y = 3x + 4. Write down the gradient of the parallel line.

Worked solution

  1. Compare with y=mx+cy = mx + c

    y=3x+4y = 3x + 4

    In the form y = mx + c the number multiplying x is the gradient.

  2. Write down the gradient of the given line

    m=3m = 3

    The coefficient of x is 3, so the given line has gradient 3.

  3. Use the rule for parallel lines

    mparallel=3m_{\text{parallel}} = 3

    Parallel lines never meet, so they have exactly the same gradient.

Answer
33
Question 2
1 markeasy
Which one of these lines is parallel to the line y=5x−1y = 5x - 1?

Worked solution

  1. Find the gradient of the given line

    y=5x−1⇒m=5y = 5x - 1 \Rightarrow m = 5

    The number in front of x is the gradient, so m=5m = 5.

  2. Use the rule for parallel lines

    mparallel=5m_{\text{parallel}} = 5

    A parallel line must have exactly the same gradient, 5.

  3. Pick the line with gradient 5

    y=5x+3y = 5x + 3

    Only y=5x+3y = 5x + 3 has gradient 5; a different intercept is fine, it just moves the line.

Answer
y=5x+3y = 5x + 3
Question 3
1 markeasy
A line is parallel to the line y=−2x+7y = -2x + 7. Write down its gradient.

Worked solution

  1. Compare with y=mx+cy = mx + c

    y=−2x+7y = -2x + 7

    In the form y = mx + c the number multiplying x is the gradient.

  2. Write down the gradient of the given line

    m=−2m = -2

    The coefficient of x is -2, so the given line has gradient -2.

  3. Use the rule for parallel lines

    mparallel=−2m_{\text{parallel}} = -2

    Parallel lines never meet, so they have exactly the same gradient.

Answer
−2-2
Question 4
1 markeasy
Work out the gradient of a line that is perpendicular to the line y=4x+1y = 4x + 1.

Worked solution

  1. Compare with y=mx+cy = mx + c

    y=4x+1y = 4x + 1

    The number multiplying x is the gradient of the given line.

  2. Use the perpendicular rule

    m1×m2=−1m_1 \times m_2 = -1

    Perpendicular gradients multiply to give -1, so m2 is the negative reciprocal of m1.

  3. Work out the negative reciprocal

    m2=−14=−14m_2 = \frac{-1}{4} = -\frac{1}{4}

    Turning 4 upside down and changing the sign gives −14-\frac{1}{4}.

Answer
−14-\frac{1}{4}
Question 5
2 markseasy
A line has equation y=−13x+2y = -\frac{1}{3}x + 2. Work out the gradient of a line perpendicular to it.

Worked solution

  1. Compare with y=mx+cy = mx + c

    y=−13x+2y = -\frac{1}{3}x + 2

    The number multiplying x is the gradient of the given line.

  2. Use the perpendicular rule

    m1×m2=−1m_1 \times m_2 = -1

    Perpendicular gradients multiply to give -1, so m2 is the negative reciprocal of m1.

  3. Work out the negative reciprocal

    m2=−1−13=3m_2 = \frac{-1}{-\frac{1}{3}} = 3

    Turning −13-\frac{1}{3} upside down and changing the sign gives 3.

Answer
33

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