Linear Equations

What is a Linear Equation?

A linear equation produces a straight line when graphed. The standard form used in the UK is:

y = mx + c

• m = gradient (slope) — how steep the line is
• c = y-intercept — where the line crosses the y-axis

In the US this is sometimes written y = mx + b, where b is the y-intercept. The letter changes, but the meaning is the same.

Key Terms

• Gradient (slope) — the rate of change; how much y changes per unit increase in x
• y-intercept — the point where the line crosses the y-axis (when x = 0)
• x-intercept — the point where the line crosses the x-axis (when y = 0)
• Linear — having a constant rate of change; always produces a straight line on a graph

Gradient (Slope)

The gradient tells you how much y changes for each unit increase in x.

m = (y₂ - y₁) / (x₂ - x₁)

Worked Example: Find the gradient through (1, 3) and (3, 7): m = (7 - 3) / (3 - 1) = 4 / 2 = 2

Interpreting gradients:

• Positive gradient → line rises from left to right
• Negative gradient → line falls from left to right
• Zero gradient → horizontal line (y = constant)
• Undefined gradient → vertical line (x = constant)

A gradient of 3 means: for every 1 unit you move right, you move up 3 units.

The y-Intercept

The y-intercept c is where the line crosses the vertical axis. To find it, set x = 0 and evaluate:

For y = 4x - 7: when x = 0, y = 4(0) - 7 = -7. The y-intercept is (0, -7).

Graphing a Line

From y = 2x + 1:

1. Plot the y-intercept: (0, 1)
2. Use the gradient (rise 2, run 1) to find another point: (1, 3)
3. Draw a straight line through both points

Worked Example: Graph y = -x + 4

• y-intercept: (0, 4)
• Gradient -1: move right 1, down 1 to get point (1, 3)
• Draw the line — it slopes downward from left to right

Finding the x-Intercept

The x-intercept is where y = 0. Set y = 0 and solve for x:

For y = 2x - 6: 0 = 2x - 6 → 2x = 6 → x = 3. The x-intercept is (3, 0).

Solving a Linear Equation for x

When given a y-value, substitute and solve step by step:

Worked Example: If y = 3x + 4 and y = 19, find x.

• 3x + 4 = 19
• 3x = 15
• x = 5

Always undo addition or subtraction first, then undo multiplication or division.

Checking if a Point is on a Line

Substitute both coordinates into the equation and check if both sides are equal:

Does (2, 5) lie on y = 2x + 1? 2(2) + 1 = 4 + 1 = 5 ✓ Yes, (2, 5) is on the line.

Does (3, 4) lie on y = 2x + 1? 2(3) + 1 = 7 ≠ 4. No, (3, 4) is not on the line.

Writing the Equation of a Line

If you know the gradient m and a point (x₁, y₁), use point-slope form: y - y₁ = m(x - x₁)

Worked Example: Gradient 3, passes through (2, 5):

• y - 5 = 3(x - 2)
• y - 5 = 3x - 6
• y = 3x - 1

Parallel and Perpendicular Lines

Parallel lines never intersect — they have the same gradient but different y-intercepts:

• y = 3x + 1 and y = 3x - 4 are parallel (both have m = 3)

Perpendicular lines meet at a right angle (90°). Their gradients are negative reciprocals:

• m₁ × m₂ = -1
• If m₁ = 2, then m₂ = -1/2
• y = 2x + 3 and y = -1/2 x + 1 are perpendicular

Common Mistakes

• Mistake: swapping m and c when reading y = mx + c. Fix: m always multiplies x; c is the standalone constant.
• Mistake: computing gradient as (x₂ - x₁) / (y₂ - y₁). Fix: it is always change in y divided by change in x (rise over run).
• Mistake: forgetting that a negative gradient makes the line fall, not rise.
• Mistake: plotting only one point and drawing the line at a guess angle. Fix: always find at least two points before drawing.

Tips and Tricks

• "Gradient" (UK) and "slope" (US) mean exactly the same thing.
• If the gradient is a fraction like 3/4, move right 4 and up 3 to find the next point accurately.
• Using a table of values is a reliable method: pick three x-values, calculate each y-value, then plot.
• Two lines with equal gradients but different y-intercepts will never meet — they are parallel.