EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
More linear graphs
Graphs · Lesson 2 of 8
Warm-up
Answer each one, then check.
1. What is the gradient of y = 5x − 2?
5
2. Work out (9 − 3)/(4 − 1).
2
3. Solve 7 = 2 × 3 + c.
c = 1
4. What is the equation of the y-axis?
x = 0
5. Work out −1 ÷ 2.
−½
Learning Objectives
1. Find the equation of a line through two points.
2. Find the equation of a line parallel to a given line.
3. Find the equation of a perpendicular line (Higher).
4. Find where two lines meet.
Line Through Two Points
Gradient first, then the y-intercept.
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1 Find the gradient m = (y₂ − y₁)/(x₂ − x₁). |
2 Write y = mx + c Put in the value of m. |
3 Substitute one point Solve for c. |
4 Write the equation And check with the other point. |
Through Two Points
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Find the equation of the line through (1, 3) and (4, 12). |
1. Gradient
(12 − 3)/(4 − 1) = 9/3 = 3
2. Substitute (1, 3) into y = 3x + c
3 = 3 + c
3. Solve for c
c = 0
4. Write the equation
y = 3x
Answer: y = 3x
Parallel and Perpendicular Lines
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Parallel lines have equal gradients. |
Two parallel lines with gradient 2 and a third line with gradient minus one half crossing them at right angles.
Parallel and Perpendicular
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PARALLEL LINES |
PERPENDICULAR LINES (HIGHER) |
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▸ They never meet. ▸ They have the same gradient. ▸ y = 2x + 1 and y = 2x − 3 are parallel. |
▸ They meet at a right angle. ▸ The gradients multiply to give −1: m₁ × m₂ = −1. ▸ y = 2x + 1 and y = −½x + 4 are perpendicular. |
A Parallel Line
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Find the equation of the line parallel to y = 2x + 3 that passes through (1, 7). |
1. Parallel lines have the same gradient
m = 2
2. Substitute (1, 7) into y = 2x + c
7 = 2 + c
3. Solve
c = 5
Answer: y = 2x + 5
A Perpendicular Line HIGHER
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Find the equation of the line perpendicular to y = 2x + 1 that passes through (4, 3). |
1. The gradient of the original is 2, so the perpendicular gradient is
−½
2. Substitute (4, 3) into y = −½x + c
3 = −2 + c
3. Solve
c = 5
Answer: y = −½x + 5
Where Two Lines Meet
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Find the coordinates of the point where y = x + 1 and y = −x + 5 cross. |
1. At the crossing point, the y-values are equal
x + 1 = −x + 5
2. Solve
2x = 4, so x = 2
3. Substitute to find y
y = 2 + 1 = 3
Answer: (2, 3)
Finding the Perpendicular Gradient HIGHER
Flip the fraction and change the sign.
|
Gradient |
Perpendicular gradient |
|---|---|
|
2 |
−½ |
|
¾ |
−4/3 |
|
−5 |
⅕ |
|
−2/7 |
7/2 |
Key Terms
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Parallel Lines that never meet, with equal gradients. |
Perpendicular Lines that meet at a right angle. |
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Intersection The point where two lines cross. |
Reciprocal A number turned upside down: the reciprocal of 2 is ½. |
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Simultaneous Happening together; the intersection satisfies both equations. |
Coordinates A pair (x, y) giving a position. |
Your Task: Which Lines Are Parallel?
10 minutes
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Sort these lines into parallel pairs and find one line perpendicular to y = 2x: y = 3x + 1, y = 3x − 4, 2y = x + 6, y = ½x − 1, y = −½x + 3. 1. Rearrange each equation. 2. Compare gradients. |
A good answer shows: y = 3x + 1 and y = 3x − 4 are parallel. 2y = x + 6 is y = ½x + 3, which is parallel to y = ½x − 1. y = −½x + 3 is perpendicular to y = 2x.
Can I...?
☐ Find the gradient from two points.
☐ Find the equation through two points.
☐ Write a line parallel to another.
☐ Recognise parallel lines from their equations.
☐ Find the perpendicular gradient (Higher).
☐ Find a perpendicular line through a point (Higher).
☐ Find where two lines cross.
☐ Check my equation with the other point.
Summary
✓ Two points: find the gradient, then substitute one point to find c.
✓ Parallel lines have equal gradients.
✓ Perpendicular gradients multiply to −1 (Higher).
✓ Where lines cross: set the y-expressions equal.
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EXAM FOCUS Find the equation of the line that is parallel to y = 3x − 2 and passes through the point (2, 9). (3 marks) Parallel means the same gradient, so write y = 3x + c and substitute the point to find c. |