How to Solve: Straight Line Graphs (y=mx+c, Parallel, Perpendicular)

How to Solve: Straight Line Graphs (y=mx+c, Parallel, Perpendicular)

Introduction

"Mastering straight-line graphs unlocks 10–15% of your GCSE Maths exam marks—from plotting lines to finding equations of parallel and perpendicular slopes. Miss this, and you’re leaving easy marks on the table."

What You Need To Know First

1. Coordinates (x, y): How to plot points on a grid.
2. Gradient (slope): How steep a line is, calculated as rise over run.
3. Substitution: Plugging numbers into equations (e.g., y = 2x + 1).

Key Vocabulary

Formulas To Know

1. Equation of a straight line (slope-intercept form):
   y = mx + c
2. m = gradient (slope)

3. c = y-intercept
   MEMORISE THIS

4. Gradient between two points (x₁, y₁) and (x₂, y₂):
   m = (y₂ – y₁) / (x₂ – x₁)
   MEMORISE THIS

5. Parallel lines:
   m₁ = m₂ (same gradient)
   MEMORISE THIS

6. Perpendicular lines:
   m₁ × m₂ = -1 (gradients multiply to -1)
   MEMORISE THIS

7. Equation from one point and gradient:
   y – y₁ = m(x – x₁)
   MEMORISE THIS (or derive from y = mx + c)

8. General form of a line:
   ax + by + c = 0
   Given on exam sheet (but know how to convert to y = mx + c)

Step-by-Step Method

How to Find the Equation of a Straight Line (y = mx + c)

Step 1: Identify the gradient (m). - If given two points, use m = (y₂ – y₁) / (x₂ – x₁). - If given a parallel line, use the same m. - If given a perpendicular line, use m₂ = -1/m₁.

Step 2: Find the y-intercept (c). - Substitute m and one point (x, y) into y = mx + c and solve for c.

Step 3: Write the equation in the form y = mx + c.

Step 4: (Optional) Convert to ax + by + c = 0 if required.

Worked Example (Using Steps Above)

Question: Find the equation of the line passing through (2, 5) and (4, 9).

Step 1: Find m. m = (9 – 5) / (4 – 2) = 4 / 2 = 2

Step 2: Find c. Use point (2, 5): 5 = 2(2) + c → 5 = 4 + c → c = 1

Step 3: Write the equation. y = 2x + 1

Step 4: (Optional) Convert to general form. 2x – y + 1 = 0

Worked Examples

Example 1 – Basic: Find the Equation from Two Points

Question: A line passes through (1, 3) and (3, 7). Find its equation.

Step 1: Gradient (m). m = (7 – 3) / (3 – 1) = 4 / 2 = 2

Step 2: Y-intercept (c). Use (1, 3): 3 = 2(1) + c → c = 1

Step 3: Equation. y = 2x + 1

What we did and why: - Found m using two points (rise over run). - Substituted one point to find c. - Wrote the final equation in y = mx + c form.

Example 2 – Medium: Parallel Line

Question: Find the equation of the line parallel to y = 3x – 2 that passes through (4, 10).

Step 1: Gradient (m). Parallel lines have the same gradient → m = 3.

Step 2: Y-intercept (c). Use (4, 10): 10 = 3(4) + c → 10 = 12 + c → c = -2

Step 3: Equation. y = 3x – 2

What we did and why: - Parallel lines share the same m, so we kept m = 3. - Used the given point to find c. - Wrote the equation in the same form.

Example 3 – Exam-Style: Perpendicular Line

Question: A line L has equation 2x + 3y – 6 = 0. Find the equation of the line perpendicular to L that passes through (6, -1).

Step 1: Convert L to y = mx + c form. 3y = -2x + 6 → y = (-2/3)x + 2 Gradient of L = -2/3

Step 2: Find perpendicular gradient. m₁ × m₂ = -1 → (-2/3) × m₂ = -1 → m₂ = 3/2

Step 3: Find c using (6, -1). -1 = (3/2)(6) + c → -1 = 9 + c → c = -10

Step 4: Write the equation. y = (3/2)x – 10

What we did and why: - Converted to y = mx + c to find m. - Used the perpendicular rule (m₁ × m₂ = -1). - Substituted the point to find c. - Wrote the final equation.

Common Mistakes

Exam Traps

1-Minute Recap (Night Before the Exam)

"Right, listen up—this is your 60-second survival guide for straight-line graphs. First, y = mx + c: m is the gradient (rise over run), c is where it hits the y-axis. If you’ve got two points, find m first, then plug in one point to get c. Parallel lines? Same m. Perpendicular? Flip m and change the sign—m₁ × m₂ = -1. If the equation’s in ax + by + c = 0, rearrange it to y = … first. And watch out for sneaky perpendicular questions—they won’t always say ‘perpendicular’! Now go smash those exam questions."