How to use this lesson
1. Complete the warm-up questions from memory
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
2. Study the worked examples carefully
3. Try the practice problems: Green → Amber → Red
4. Check your answers and complete the mastery checklist
🔥
Warm-Up: Retrieval Practice⏱ Before You Start
Answer from memory — no notes!Q1 Find gradient from (0, 0) to (3, 9)
Q2 Identify y-intercept in: y = 4x − 7
Q3 Write equation: gradient 2, intercept 5
Q4 🔄 RECALL Solve: −2x > 8
Q4 Solve: −2x > 8
Lesson Progress
✓ Warm-up — complete
→ Worked examples & key concepts
· Tiered practice (Green → Amber → Red)
· Mistake Detective & Examiner Lens
· Mastery checklist
💡
The Big Idea: Lines Have RulesEvery straight line can be described by y = mx + c
m = gradient
c = y-intercept
In coordinate geometry, we find m and c from given information: points, gradients, or parallel lines.
Gradient = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁)
📖
Core DefinitionsGradient (m)
Change in y ÷ change in x (rise over run)
y-intercept (c)
Value of y when x = 0 (where line crosses y-axis)
Parallel Lines
Lines with the same gradient
Point Substitution
Replacing x and y with coordinates to find c
Midpoint
The point exactly halfway between two coordinates — average the x's, average the y's
✏️
Worked ExamplesWorked Example 01Given Gradient and Intercept
Gradient = 3, y-intercept = 4
1
Substitute into
y = mx + c: m = 3, c = 4✓ Answer
y = 3x + 4
Worked Example 02Gradient and Point on y-axis
Gradient = 2, passing through (0, 5)
1
Point where x = 0 gives intercept directly:
c = 5✓ Answer
y = 2x + 5
Worked Example 03Gradient and Non-Zero PointKEY SKILL
Gradient = 4, passing through (2, 3)
1
Start with:
y = 4x + c2
Substitute (2, 3):
3 = 4(2) + c → 3 = 8 + c3
Solve:
c = 3 − 8 = −5✓ Answer
y = 4x − 5
Worked Example 04Given Two PointsNEW
Find equation through (1, 3) and (5, 11)
1
Gradient: rise = 11 − 3 = 8, run = 5 − 1 = 4 →
m = 8 ÷ 4 = 22
Start:
y = 2x + c3
Substitute (1, 3):
3 = 2(1) + c → c = 1✓ Answer
y = 2x + 1
Worked Example 05Parallel Lines
Find equation parallel to y = 3x − 2, through (0, 7)
1
Parallel = same gradient →
m = 32
Point (0, 7) gives
c = 7✓ Answer
y = 3x + 7
💡 Parallel lines: same gradient, different y-intercept.
Parallel lines never meet — they always have the same gradient but different y-intercepts.
🔍 Upper Foundation Boundary
Perpendicular Lines: gradients multiply to −1. If one gradient is 2, the perpendicular gradient is −½. This appears occasionally on higher Foundation papers.Worked Example 06Finding the MidpointNEW
Find the midpoint of (2, 3) and (8, 11)
1
Average the x-coordinates:
(2 + 8) ÷ 2 = 52
Average the y-coordinates:
(3 + 11) ÷ 2 = 7✓ Midpoint
(5, 7)
💡 Midpoint = "average the x's, average the y's". It's the point exactly halfway between.
The midpoint M is the average of both coordinates — exactly halfway between A and B.
✅
Process Checklist📋 Finding the Equation of a Line
✓
Identify gradient (m)✓
Identify intercept (c) — or find it by substitution✓
If c unknown: substitute a point into y = mx + c✓
Solve for c✓
Write final answer in form y = mx + c🧮
Arithmetic SupportRise = y₂ − y₁
Run = x₂ − x₁
Substitute carefully
Keep subtraction order consistent
💪
Tiered Practice🔓 How It Works
Tap any card to reveal the full working and answer.0 / 14
Problems revealed
Tap problems to reveal working and answers
🟢 Green — Core Skills
🟢 Green · TYPE YOUR ANSWER
Find gradient: (2, 1) to (6, 9)
✍️ Type Your Answer
Find the gradient between (1, 2) and (3, 8).
Gradient = (8−2)/(3−1) = 6/2 = 3
✓ 3
Green · Q1
Write equation: gradient 5, intercept −3
m = 5, c = −3
Use y = mx + c
✓ y = 5x − 3
Tap to reveal answer
Green · Q2
Equation through (0, 6) with gradient 2
x = 0 gives c directly
c = 6, m = 2
✓ y = 2x + 6
Tap to reveal answer
Green · Q3
Find gradient: (2, 4) to (6, 12)
Rise = 12 − 4 = 8
Run = 6 − 2 = 4
m = 8 ÷ 4
✓ m = 2
Tap to reveal answer
🟡 Amber — Multi-Step
🟡 Amber · TYPE YOUR ANSWER
Write equation: gradient 4, y-intercept −2
Amber · Q4
Equation: gradient 3, through (1, 2)
y = 3x + c
2 = 3(1) + c
2 = 3 + c → c = −1
✓ y = 3x − 1
Tap to reveal answer
Amber · Q5
Through (2, −1) and (6, 7)
m = (7−(−1))÷(6−2) = 8÷4 = 2
y = 2x + c
−1 = 2(2) + c → c = −5
✓ y = 2x − 5
Tap to reveal answer
Amber · Q6
Parallel to y = 4x + 1, through (0, −5)
Parallel → same gradient
m = 4, c = −5
✓ y = 4x − 5
Tap to reveal answer
Amber · Q7
Gradient −2, through (3, 1)
y = −2x + c
1 = −2(3) + c
1 = −6 + c → c = 7
✓ y = −2x + 7
Tap to reveal answer
🔴 Red — Exam Style
🔴 Red · TYPE YOUR ANSWER
Find the midpoint of (3, 5) and (7, 11)
Red · 3 marks · Q8
Line through (2, 3) and (8, 15). Find equation
m = (15−3)÷(8−2) = 12÷6 = 2
y = 2x + c
3 = 2(2) + c → c = −1
✓ y = 2x − 1
Tap to reveal answer
Red · 3 marks · Q9
Gradient −3, through (4, 5). Find equation
y = −3x + c
5 = −3(4) + c
5 = −12 + c → c = 17
✓ y = −3x + 17
Tap to reveal answer
Red · 3 marks · Q10
Parallel to y = −2x + 1, through (3, 4)
Parallel → m = −2
y = −2x + c
4 = −2(3) + c → 4 = −6 + c → c = 10
✓ y = −2x + 10
Tap to reveal answer
🟡 Midpoint Questions
Amber · Q11
Find midpoint of (0, 0) and (6, 4)
x: (0+6)÷2 = 3
y: (0+4)÷2 = 2
✓ (3, 2)
Tap to reveal answer
Amber · Q12
Find midpoint of (−2, 5) and (4, 1)
x: (−2+4)÷2 = 1
y: (5+1)÷2 = 3
✓ (1, 3)
Tap to reveal answer
Red · 2 marks · Q13
Find midpoint of (3, −4) and (7, 8)
x: (3+7)÷2 = 5
y: (−4+8)÷2 = 2
✓ (5, 2)
Tap to reveal answer
🔍
Mistake Detective⚠️ These mistakes cost marks
Study each one carefully.❌ Wrong
Gradient = (x₂ − x₁) ÷ (y₂ − y₁)
Rise and run are reversed!
✅ Correct
m = (y₂ − y₁) ÷ (x₂ − x₁)
Rise over run
❌ Wrong
Forgot to substitute point to find c
y = 2x + ???
Can’t finish the equation without finding c
✅ Correct
Substitute (x, y) into y = mx + c
Then solve for c
e.g. 3 = 2(1) + c → c = 1
🎓
Examiner Lens🎓 To Gain Full Marks
- Show gradient calculation clearly (rise ÷ run)
- Show substitution step when finding c
- Solve for c clearly with working
- Write final equation in correct form y = mx + c
- For parallel lines, state “same gradient” explicitly
🏆
Mastery Checklist🏆 Unit 10 Mastery
✓
I can find gradient between two points
✓
I can find equation given gradient and intercept
✓
I can find equation given gradient and a point (substitution)
✓
I can find equation given two points
✓
I understand parallel lines have the same gradient
✓
I can find the midpoint of two coordinates
🎉 Congratulations — you've completed all 10 units!
You've worked through the entire GCSE Maths Rescue System. That's serious commitment. Go into your exam with confidence.
Exam Ready?
If you can do these 3 things confidently, you're ready to move on:
Your Grade 3→5 Rescue System
10 units · Algebra, Graphs & Coordinate Geometry
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✔ Step-by-step explanations
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