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Parallel and perpendicular lines
Gradients tell you instantly whether two lines are parallel or meet at right angles.
Parallel lines
Parallel lines have the same gradient.
y = 3x + 1 and y = 3x − 5 are parallel.
y = 3x + 1 and y = 3x − 5 are parallel.
Perpendicular lines
Perpendicular gradients multiply to −1.
If one gradient is m, the other is −1m: flip it and change the sign.
Gradient 2 → perpendicular gradient −12.
If one gradient is m, the other is −1m: flip it and change the sign.
Gradient 2 → perpendicular gradient −12.
Finding the equation
Find the new gradient, then substitute the point to find c.
Perpendicular to y = 2x + 1 through (4, 3): m = −12. 3 = −2 + c, so c = 5.
y = −12x + 5
Perpendicular to y = 2x + 1 through (4, 3): m = −12. 3 = −2 + c, so c = 5.
y = −12x + 5
Find the equation of the line parallel to y = 4x − 1 through (2, 11).
- Same gradient: m = 4.
- 11 = 4 × 2 + c, so c = 3.
Answer: y = 4x + 3
Parallel lines share a gradient. Perpendicular gradients multiply to −1, so flip and change the sign. Then substitute a point to find c.
Check you have got it
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