Use the equation of a straight line to solve problems involving gradients and intercepts

Straight‑Line Graphs

What is a straight line?

A straight line on a graph is described by the simple equation $$y = mx + c$$ where m is the gradient (slope) and c is the y‑intercept (the point where the line crosses the y‑axis). Think of m as the “steepness” of a hill you’re walking up: a higher m means a steeper climb. The y‑intercept is where you start on the y‑axis before you begin walking.

Finding the Gradient (m)

The gradient is the change in y divided by the change in x between any two points on the line: $$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$ Imagine you’re measuring how fast your speed increases while driving: the gradient tells you how many miles you gain for every mile you travel forward.

1. Pick two points on the line, e.g. (2, 5) and (5, 11).
2. Compute Δy = 11 − 5 = 6.
3. Compute Δx = 5 − 2 = 3.
4. Gradient m = 6 ÷ 3 = 2.

Finding the y‑Intercept (c)

Once you know m, substitute any point into the equation to solve for c: $$c = y - mx$$ Using the point (2, 5) and gradient 2: $$c = 5 - 2 \times 2 = 5 - 4 = 1$$ So the line’s equation is y = 2x + 1.

Graphing the Line

1. Plot the y‑intercept (0, 1). 2. Use the gradient: rise 2 for every run 1. 3. Mark another point: from (0, 1) move right 1 to (1, 0) and up 2 to (1, 3). 4. Draw a straight line through these points.

Examples & Analogies

• 🚗 Speed vs. Time: If a car travels at a constant speed of 60 km/h, the graph of distance (y) against time (x) is a straight line with gradient 60.
• 📈 Profit vs. Units Sold: Profit increases by a fixed amount for each additional unit sold, giving a straight line with a positive gradient.
• 📉 Depreciation: A laptop losing value at a steady rate will have a straight line with a negative gradient.

Practice Problems

1. Given points (3, 7) and (6, 13), find the gradient and write the equation of the line.
2. A line has equation y = -4x + 8. What is its gradient and y‑intercept?
3. Draw the graph of y = 0.5x - 2 on a coordinate plane.

Key Terms

Quick Check

- If m = 0, the line is horizontal. - If m is undefined (vertical line), the equation is x = k. - The product of slopes of two perpendicular lines is -1.