Equations of straight lines

What you will learn

• 10.1 Find the equation of a line from:
• 10.1 Read the features of a line from its equation: slope, intercepts, whether it passes through a point. Slope and inclination (slope = tan θ). The normal form is not required.
• 10.2 Number of intersection points of two lines (0, 1, or infinitely many) from their equations.

You need first: Coordinates, Number and algebra basics

Key terms

Standard steps

Find the equation

1. Get the slope m (given, or from two points: m = (y₂ − y₁)/(x₂ − x₁); parallel → same m; perpendicular → −1/m).
2. Get one point (x₁, y₁) on the line.
3. Point-slope form: y − y₁ = m(x − x₁).
4. Tidy into the form the question wants:
   • y = mx + c, or
   • general form Ax + By + C = 0, usually with integer coefficients and A > 0.

Read the line Ax + By + C = 0

Safer: put x = 0 and y = 0 yourself rather than memorising.

Two lines

1. Compare slopes:
   • different slopes → exactly one intersection;
   • same slope, different intercepts → parallel, no intersection;
   • same line → infinitely many.
2. To find the intersection, solve the two equations simultaneously.

Does a point lie on the line?

1. Substitute the point's x and y into the equation.
2. Work out both sides.
3. If they are equal, the point lies on the line; if not, it does not.

Worked examples

y − (−1) = 3(x − 2)
y + 1 = 3x − 6
y = 3x − 7

m = (−4 − 4)/(3 − (−1)) = −8/4 = −2
y − 4 = −2(x + 1)
y = −2x + 2

slope = −3/(−4) = 3/4
x = 0: −4y + 12 = 0, y-intercept 3
y = 0: 3x + 12 = 0, x-intercept −4

slope of 2x + y − 5 = 0 is −2, so slope of L = 1/2
y − 1 = ½(x − 4)
2y − 2 = x − 4
L: x − 2y − 2 = 0

solve 2x + y = 5 and x − 2y = 2:
x = 12/5, y = 1/5      intersection (12/5, 1/5)

(2, 3):  3(2) − 4(3) + 6 = 6 − 12 + 6 = 0     ✓ lies on the line
(1, 2):  3(1) − 4(2) + 6 = 3 − 8 + 6 = 1 ≠ 0  ✗ does not

Common mistakes

• Using the slope of a line as the slope of the line perpendicular to it.
• Slope of Ax + By + C = 0 written as A/B (the sign is wrong).
• Leaving fractions in the general form; multiply through.
• Mixing up the x- and y-intercepts.
• Treating "same slope" as "same line".

How the DSE tests it

• A(2): the equation of a perpendicular bisector (a locus) and the area of a figure.
• Section B: lines combined with circles, triangle centres, or a moving point.
• Paper 2: match a line's equation to its graph, using the signs of slope and intercepts.

Questions you might not ask in class

More Form 4 maths topics