EDEXCEL GCSE MATHS · FOUNDATION & HIGHER
More linear graphs
Graphs · Lesson 2 of 8
Last Lesson and Before
Answer each one, then check.
1. Last lesson: what is the gradient of y = 7 − 2x?
−2
2. Last lesson: gradient between (0, 1) and (2, 9)?
4
3. Solve 5 = 3 × 2 + c.
c = −1
4. What is the reciprocal of 4?
¼
Learning Objectives
1. Find the equation of a line from its gradient and one point.
2. Find the equation of a line through two points.
3. Recognise and find parallel lines.
4. Draw a line given in the form ax + by = c.
5. (Higher) Recognise and find perpendicular lines.
Parallel Lines
Parallel lines have the same gradient.
▸ Same m. y = 3x + 1, y = 3x − 4 and y = 3x are all parallel.
▸ Different c. Parallel lines cross the y-axis at different points.
▸ Rearrange to compare. 2y = 6x + 5 is y = 3x + 2.5, so it is parallel to y = 3x + 1.
▸ Same line. If m and c are both the same, it is the same line.
Equation from a Point and a Gradient
Write y = mx + c, then use the point to find c.
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1 Gradient Write down m - given, or from a parallel line, or worked out from two points. |
2 Substitute Put the point's x and y into y = mx + c. |
3 Solve Solve for c. |
4 Write Write the equation in full: y = mx + c with numbers. |
A Parallel Line through a Point
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Find the equation of the line parallel to y = 3x − 4 that passes through (2, 5). |
1. Parallel, so the same gradient
m = 3, so y = 3x + c
2. Substitute (2, 5)
5 = 3 × 2 + c
3. Solve
c = −1
Answer: y = 3x − 1
A Line through Two Points
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Find the equation of the line through (1, 4) and (3, 10). |
1. Gradient
(10 − 4)/(3 − 1) = 6/2 = 3
2. Substitute (1, 4) into y = 3x + c
4 = 3 + c, so c = 1
3. Check with the other point
3 × 3 + 1 = 10 - correct
Answer: y = 3x + 1
Drawing 3x + 2y = 12
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Find where 3x + 2y = 12 crosses the axes, and find its gradient. |
1. Crosses the y-axis when x = 0
2y = 12, so (0, 6)
2. Crosses the x-axis when y = 0
3x = 12, so (4, 0)
3. Rearrange for the gradient
2y = −3x + 12, so y = −1.5x + 6
Answer: Through (0, 6) and (4, 0); gradient −1.5
PART TWO · HIGHER
Perpendicular Lines
Lines that meet at right angles.
Parallel and Perpendicular
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If one line has gradient m, a line perpendicular to it has gradient −1/m, the negative reciprocal. Here 2 × (−½) = −1. |
Parallel: equal gradients. Perpendicular: gradients multiply to −1. |
Perpendicular Gradients
Flip the gradient and change its sign.
▸ Negative reciprocal. Perpendicular to gradient m is gradient −1/m.
▸ Examples. 2 goes to −½; −3 goes to ⅓; 2/5 goes to −5/2.
▸ The test. Two lines are perpendicular if m₁ × m₂ = −1.
▸ Then as before. Substitute the point to find c.
A Perpendicular Line through a Point
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Find the equation of the line perpendicular to y = 2x + 3 that passes through (4, 1). |
1. Perpendicular gradient
m = −½
2. Substitute (4, 1)
1 = −½ × 4 + c = −2 + c
3. Solve
c = 3
Answer: y = −½x + 3
Key Terms
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Parallel Lines with the same gradient that never meet. |
Perpendicular Lines that meet at a right angle. |
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Negative reciprocal −1/m: the gradient of a line perpendicular to one with gradient m. |
Intercept Where a line crosses an axis. |
Your Task: Build a Square
12 minutes
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The line y = 2x passes through (0, 0) and (1, 2). Find the equations of three more lines that make a square with it, with (0, 0) and (1, 2) as two neighbouring corners. Check your square on squared paper. 1. Parallel sides share a gradient. 2. Neighbouring sides are perpendicular. 3. Check each corner lies on two lines. |
A good answer shows: Sides through the origin and (1, 2) must be perpendicular to y = 2x: y = −½x and y = −½x + 2.5. The fourth side is parallel to y = 2x through (−2, 1): y = 2x + 5. (The square on the other side gives y = 2x − 5 instead.)
Can I...?
☐ Find the equation of a line from a point and a gradient.
☐ Find the equation of a line through two points.
☐ Tell whether lines are parallel.
☐ Draw a line in the form ax + by = c.
☐ (Higher) Find a perpendicular gradient.
☐ (Higher) Find the equation of a perpendicular line.
Summary
✓ Parallel lines: same gradient.
✓ Find c by substituting a point into y = mx + c.
✓ ax + by = c: set x = 0 then y = 0 to find where it crosses the axes.
✓ (Higher) Perpendicular gradients multiply to −1.
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EXAM FOCUS Find the equation of the line that is parallel to y = 3x − 4 and passes through (2, 5). (3 marks) Always check your final equation by substituting the point back in. If the y doesn't come out right, you've made a slip finding c. |