GCSE Maths: Parallel and perpendicular lines (Foundation)
Part of the GCSE Maths straight-line graphs and coordinates chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Parallel and perpendicular lines
Parallel lines go in the same direction and never meet, like the two rails of a straight railway track. They have exactly the same steepness, so parallel lines have equal gradients. In y = mx + c form, the lines y = 2x + 1, y = 2x − 3 and y = 2x are all parallel because every one has m = 2. Only the y-intercept (c) is different, which just slides the line up or down.
If an equation is not in the form y = mx + c, rearrange it first before comparing gradients. For example 2y = 4x + 6 becomes y = 2x + 3, so it is parallel to y = 2x + 1.
Finding a line parallel to another
To find the equation of a line parallel to y = 2x + 1 that passes through (3, 4): the gradient must be 2, so write y = 2x + c, substitute the point to get 4 = 6 + c, so c = −2. The line is y = 2x − 2. Keep the gradient, find a new c.
Key ideas
- Parallel lines have equal gradients. Rearrange to y = mx + c to compare.
- A parallel line through a point: keep m, substitute the point to find the new c.
- Higher: perpendicular gradients multiply to −1: the perpendicular gradient is the negative reciprocal (flip and change sign).
- Higher: perpendicular bisector = through the midpoint, with the perpendicular gradient.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Rearrange the second equation into the form y = mx + c, then compare the gradients.
- Find the gradient of the first line. The equation is already y = mx + c.y = 3x − 4 has m = 3
- Rearrange the second equation. Divide every term by 3 so that y is on its own.3y = 9x + 2 → y = 3x + 23
- Compare the gradients. Parallel lines have the same gradient.3 = 3
- Write the conclusion. The lines are parallel.
Check. Both lines rise 3 for every 1 across. ✓
Watch out. You must rearrange first. In 3y = 9x + 2 the gradient is not 9.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 2x + 1 has gradient m = 2
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 2x + c
- Substitute the point (3, 4) to find c. The new line must go through this point.4 = 2 × 3 + c
4 = 6 + c
c = 4 − 6 = −2 - Write the equation. y = 2x − 2
Check. The gradient 2 matches the given line, and the point (3, 4) fits: 2 × 3 − 2 = 4. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = −12x + 3 has gradient m = −12
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = −12x + c
- Substitute the point (4, 5) to find c. The new line must go through this point.5 = (−12) × 4 + c
5 = −2 + c
c = 5 − (−2) = 7 - Write the equation. y = −12x + 7
Check. The gradient −12 matches the given line, and the point (4, 5) fits: (−12) × 4 + 7 = 5. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Watch out
Same gradient, not same c. Parallel lines have equal m; if c is also equal they are the same line. Always rearrange to y = mx + c first: the gradient of 2y = 6x + 1 is 3, not 6. For perpendicular lines you must flip and change the sign.Exercise 12I Parallel lines
Non-calculatorMethod. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = 3x + 2 is in the form y = mx + c.m = 3
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = 3
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = −x + 7 is in the form y = mx + c.m = −1
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = −1
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = 12x − 1 is in the form y = mx + c.m = 12
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = 12
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as the given line, so find m.
- Find the number multiplying x. y = −2x + 5 is in the form y = mx + c.m = −2
- Parallel lines have equal gradients. Same steepness, never meet.parallel gradient = −2
Check. Equal gradients mean the same slope. ✓
Watch out. Keep the sign of the gradient.
Method. Parallel lines have the same gradient as y = 2x + 1, which is 2. Check each line.
- Find the gradient of y = 2x + 1. The number multiplying x.gradient 2
- Check A to D. Write the gradient of each line.A: 2 ✓ B: 1 ✗ C: 2 ✓ D: −2 ✗
- Choose the lines with gradient 2. The y-intercepts do not matter.A and C
Check. A and C both go up 2 for each 1 across. ✓
Watch out. D has gradient −2: it goes down, so it is not parallel.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Read the y-intercept (c), find the gradient (m) with a triangle, then write y = mx + c.
- Find where the line crosses the y-axis. That value is c, the y-intercept.The line crosses the y-axis at (0, 1), so c = 1.
- Draw a gradient triangle. Choose two points on grid corners and count the run (across) and the rise (up or down).
- Work out the gradient. Gradient = rise ÷ run.rise = 3, run = 6
m = 3 ÷ 6 = 12 - Write the equation in the form y = mx + c. Put m and c into the formula.y = 12x + 1
Check. Test a point on the line, (−4, −1): y = 12 × (−4) + 1 = −1, which matches. ✓
Watch out. Do not mix up m and c: c is where the line hits the y-axis; m is the steepness. A negative gradient needs a minus sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Read the y-intercept (c), find the gradient (m) with a triangle, then write y = mx + c.
- Find where the line crosses the y-axis. That value is c, the y-intercept.The line crosses the y-axis at (0, −2), so c = −2.
- Draw a gradient triangle. Choose two points on grid corners and count the run (across) and the rise (up or down).
- Work out the gradient. Gradient = rise ÷ run.rise = 3, run = 6
m = 3 ÷ 6 = 12 - Write the equation in the form y = mx + c. Put m and c into the formula.y = 12x − 2
Check. Test a point on the line, (−4, −4): y = 12 × (−4) − 2 = −4, which matches. ✓
Watch out. Do not mix up m and c: c is where the line hits the y-axis; m is the steepness. A negative gradient needs a minus sign.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 2x + 1 has gradient m = 2
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 2x + c
- Substitute the point (3, 4) to find c. The new line must go through this point.4 = 2 × 3 + c
4 = 6 + c
c = 4 − 6 = −2 - Write the equation. y = 2x − 2
Check. The gradient 2 matches the given line, and the point (3, 4) fits: 2 × 3 − 2 = 4. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = −x + 4 has gradient m = −1
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = −x + c
- Substitute the point (2, 5) to find c. The new line must go through this point.5 = (−1) × 2 + c
5 = −2 + c
c = 5 − (−2) = 7 - Write the equation. y = −x + 7
Check. The gradient −1 matches the given line, and the point (2, 5) fits: −2 + 7 = 5. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 12x + 3 has gradient m = 12
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 12x + c
- Substitute the point (4, 0) to find c. The new line must go through this point.0 = 12 × 4 + c
0 = 2 + c
c = 0 − 2 = −2 - Write the equation. y = 12x − 2
Check. The gradient 12 matches the given line, and the point (4, 0) fits: 12 × 4 − 2 = 0. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = −3x has gradient m = −3
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = −3x + c
- Substitute the point (−1, 2) to find c. The new line must go through this point.2 = (−3) × (−1) + c
2 = 3 + c
c = 2 − 3 = −1 - Write the equation. y = −3x − 1
Check. The gradient −3 matches the given line, and the point (−1, 2) fits: (−3) × (−1) − 1 = 2. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Method. Parallel lines have the same gradient, so m = 4. The y-intercept gives c directly.
- Find the gradient. The number in front of x.m = 4
- Find c. The y-intercept is −2.c = −2
- Write the equation. y = 4x − 2
Check. When x = 0, y = −2. ✓
Watch out. Keep the sign: −2 is subtracted.
Method. Two lines are parallel if they have the same gradient.
- Write the gradient of each line. Put each equation in the form y = mx + c. The gradient is m.y = 2x + 3: gradient 2
2y = 4x − 5 (divide by 2: y = 2x − 2.5): gradient 2 - Compare the gradients. Parallel lines have equal gradients: they slope the same way and never meet.2 = 2
- Decide. The lines are parallel.
Check. Draw both lines: parallel lines never cross. ✓
Watch out. The y-intercepts may be different: only the gradients need to match.
Method. Two lines are parallel if they have the same gradient.
- Write the gradient of each line. Put each equation in the form y = mx + c. The gradient is m.4x + 2y = 7 (subtract 4x, divide by 2: y = −2x + 3.5): gradient −2
y = −2x + 1: gradient −2 - Compare the gradients. Parallel lines have equal gradients: they slope the same way and never meet.−2 = −2
- Decide. The lines are parallel.
Check. Draw both lines: parallel lines never cross. ✓
Watch out. The y-intercepts may be different: only the gradients need to match.
Method. A quadrilateral with two pairs of parallel sides is a parallelogram. Parallel sides have equal gradients, so work out all four gradients.
- Draw a sketch. The sketch shows the shape.
- Gradient of AB and of DC. Rise ÷ run for each side, in the same direction.AB: (1 − (−1)) ÷ (2 − (−2)) = 12
DC: (4 − 2) ÷ (3 − (−1)) = 12 - Gradient of BC and of AD. The other pair of sides.BC: (4 − 1) ÷ (3 − 2) = 3
AD: (2 − (−1)) ÷ (−1 − (−2)) = 3 - Conclude. Equal gradients mean parallel sides.AB is parallel to DC, and BC is parallel to AD. ABCD has two pairs of parallel sides, so it is a parallelogram.
Check. The midpoints of the diagonals AC and BD are both (0.5, 1.5). ✓
Watch out. Compare gradients of opposite sides.
Method. Parallel lines have equal gradients. Write the gradient of each line and make them equal.
- Find the gradient of the first line. y = 3x + 1 is in the form y = mx + c.m = 3
- Find the gradient of the second line. In y = kx − 5 the gradient is k.m = k
- Make them equal. The lines are parallel.k = 3
Check. y = 3x + 1 and y = 3x − 5 both have gradient 3. ✓
Watch out. Only the gradients must match.
Method. Write the gradient of the second line in terms of k, then make it equal to 10.
- Find the gradient of y = 10x. m = 10
- Find the gradient of y = 2kx + 1. The number multiplying x is 2k.m = 2k
- Make them equal and solve. Divide both sides by 2.2k = 10, so k = 5
Check. 2 × 5 = 10. ✓
Watch out. The gradient is the whole number multiplying x, here 2k.
Key words. The equation y = mx + c describes a straight line. m is the gradient (steepness) and c is the y-intercept (the value of y where the line crosses the y-axis, which is where x = 0).
Method. Parallel lines have the same gradient. Keep m, then use the point to find the new c.
- Read the gradient of the given line. The equation is in the form y = mx + c, so m is the number in front of x.y = 12x + 1 has gradient m = 12
- Use the same gradient. Parallel lines have equal gradients: they have the same steepness so they never meet.y = 12x + c
- Substitute the point (4, 6) to find c. The new line must go through this point.6 = 12 × 4 + c
6 = 2 + c
c = 6 − 2 = 4 - Write the equation. y = 12x + 4
Check. The gradient 12 matches the given line, and the point (4, 6) fits: 12 × 4 + 4 = 6. ✓
Watch out. Only the gradient stays the same. If you keep c as well you get the same line, not a parallel one.
Method. Two lines with the same gradient go in the same direction and keep the same distance apart.
- Find the gradients. Both equations are in the form y = mx + c.y = 2x + 1 has m = 2 and y = 2x + 4 has m = 2
- Think about what equal gradients mean. For every 1 across, each line goes up 2.They stay the same vertical distance apart: 4 − 1 = 3.
- Conclude. Lines that stay the same distance apart can never meet.The lines are parallel and have different y-intercepts, so they never meet.
Check. Suppose they met: 2x + 1 = 2x + 4 would give 1 = 4, which is impossible. ✓
Watch out. “Parallel” is the key word.
Method. Find the gradient of each line. Lines with the same gradient are parallel.
- Write the gradient of each line. The number multiplying x. The line y = 3 has no x term, so its gradient is 0.A: 2, B: −1, C: 2, D: −1, E: 12, F: 0
- Group equal gradients. Equal gradients are parallel.Gradient 2: A and C. Gradient −1: B and D.
- Answer. E and F have no partner.The parallel pairs are A and C, and B and D.
Check. A and C both go up 2 for 1 across. B and D both go down 1 for 1 across. ✓
Watch out. Check the sign of each gradient.