Practice Maths

L44 — Equations of Lines

Key Terms

Gradient–intercept form
y = mx + b — the most common form; m is the gradient, b is the y-intercept.
Point–gradient form
y − y1 = m(x − x1) — use when you know the gradient and one point on the line.
General form
ax + by + c = 0 where a, b, c are integers — rearranged form useful for comparing and solving systems.
x-intercept
The point where the line crosses the x-axis; found by setting y = 0 and solving for x.
y-intercept
The point where the line crosses the y-axis; found by setting x = 0; equals b in y = mx + b.
Simultaneous equations
Two or more equations solved together to find the point of intersection of the corresponding lines.

Forms of a line equation

FormEquationBest used when…
Gradient–intercept y = mx + b You know the gradient m and y-intercept b
Point–gradient y − y1 = m(x − x1) You know the gradient m and one point (x1, y1)
Two-point Calculate m first, then use point–gradient You know two points on the line
General form ax + by + c = 0 Rearranged form; a, b, c are integers

Special lines

Line typeEquationGradient
Horizontaly = k (constant)m = 0
Verticalx = h (constant)undefined
Through originy = mxm (as given)
x y 1 2 3 4 5 6 0 1 2 3 4 y = x + 1 y = −x + 3 y = 2 (1,2) (0,1) (0,3)
Three lines on the same axes. The two slanted lines intersect at (1, 2).
Hot Tip: When finding the equation through two points: always find the gradient first using m = (y2−y1)/(x2−x1), then substitute one point into y − y1 = m(x − x1). Pick the cleaner of the two points to reduce arithmetic errors.

Worked Example 1 — Gradient–intercept form

A line has gradient −2 and y-intercept 5. Write its equation and find where it crosses the x-axis.

y = −2x + 5. x-intercept: 0 = −2x + 5 ⇒ x = 5/2 = 2.5.

Worked Example 2 — Point–gradient form

Find the equation of the line with gradient 3 passing through (2, −1).

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

Worked Example 3 — Two-point form

Find the equation of the line through A(1, 4) and B(5, 12).

m = (12−4)/(5−1) = 8/4 = 2. Using A: y − 4 = 2(x − 1) ⇒ y = 2x + 2.

Check with B: y = 2(5)+2 = 12 ✓. Equation: y = 2x + 2.

Worked Example 4 — Intersection of two lines

Find the point of intersection of y = 3x − 1 and y = −x + 7.

3x − 1 = −x + 7 ⇒ 4x = 8 ⇒ x = 2. y = 3(2)−1 = 5. Intersection: (2, 5).

Worked Example 5 — Parallel and perpendicular lines through a point

The line L: y = 2x + 3. Find equations of lines through (4, 1) that are: (a) parallel; (b) perpendicular to L.

(a) m = 2: y − 1 = 2(x − 4) ⇒ y = 2x − 7.

(b) m = −1/2: y − 1 = −1/2(x − 4) ⇒ y − 1 = −x/2 + 2 ⇒ y = −x/2 + 3.

See Answers ➔

  1. Gradient–intercept form. Fluency

    • (a) Write the equation of a line with gradient 4 and y-intercept −3.
    • (b) State the gradient and y-intercept of y = −3x + 7.
    • (c) Find the x-intercept of y = 2x − 6.
    • (d) A line has equation y = 5x + 2. Find y when x = −1.

  2. Point–gradient form. Fluency

    • (a) Find the equation of a line with gradient 2 through (3, 5).
    • (b) Find the equation of a line with gradient −1 through (0, 4).
    • (c) Find the equation of a line with gradient 1/2 through (−4, 1).
    • (d) Write y − 3 = 4(x + 1) in gradient–intercept form.

  3. Two-point form. Fluency

    • (a) Find the equation of the line through A(0, 3) and B(2, 7).
    • (b) Find the equation of the line through P(1, −1) and Q(4, 5).
    • (c) Find the equation of the line through R(3, 0) and S(0, 6).
    • (d) A line passes through (2, 5) and (−1, −4). Write it in general form ax + by + c = 0.

  4. Intersection of lines. Fluency

    • (a) Find the intersection of y = x + 2 and y = −x + 6.
    • (b) Find the intersection of y = 3x − 1 and y = x + 3.
    • (c) Find the intersection of 2x + y = 8 and x − y = 1.
    • (d) Do y = 2x + 1 and y = 2x − 3 intersect? Explain.

  5. Lines on a graph. Understanding

    The graph shows three lines. Use it to answer the questions.

    x y 1 2 3 4 5 0 1 2 3 4 A B C (2,2)
    Three concurrent lines meeting at (2, 2).
    • (a) Write the equation of each line A, B, and C from the graph.
    • (b) Verify algebraically that all three lines pass through (2, 2).
    • (c) Which two lines are perpendicular? Justify using gradients.
    • (d) Write the equation of a fourth line through (2, 2) with gradient 3.

  6. Parallel and perpendicular through a point. Understanding

    • (a) Find the equation of the line through (1, 3) parallel to y = 4x − 5.
    • (b) Find the equation of the line through (0, 2) perpendicular to y = −3x + 1.
    • (c) Two lines are y = kx + 2 and y = 3x − 4. Find k if they are parallel. Find k if they are perpendicular.
    • (d) A line perpendicular to 3x − 4y = 12 passes through (3, −1). Find its equation.

  7. Lines and regions. Understanding

    • (a) Is the point (3, 5) above or below the line y = 2x − 1? Show working.
    • (b) Write the inequality that represents all points above the line y = x + 3.
    • (c) The region between y = x and y = x + 4 (for x ≥ 0). Describe this region and find its width at x = 2.
    • (d) A line has equation 2x + 3y = 12. Rearrange to gradient–intercept form and find the gradient and both intercepts.

  8. Systems of linear equations. Understanding

    • (a) Three lines are y = x + 1, y = 2x − 1, y = −x + 3. Find the vertices of the triangle they form.
    • (b) Lines L1: y = 2x + c and L2: y = −x + 6 intersect at (1, 5). Find c and verify the intersection.
    • (c) The x-intercept of a line is 4 and its y-intercept is −6. Find the gradient and write the equation.
    • (d) Two lines are parallel: y = 3x + k and 6x − 2y + 10 = 0. Find k.

  9. Applied linear modelling. Problem Solving

    A plumber charges a call-out fee of $80 plus $60 per hour. A second plumber charges $50 per hour with no call-out fee.

    • (a) Write cost functions C1(h) and C2(h) for each plumber.
    • (b) For how many hours are the costs equal?
    • (c) Graph both lines on the same axes (sketch). Label the intersection.
    • (d) For a 5-hour job, which plumber is cheaper? For a 1-hour job?

  10. Perpendicular bisector and triangle properties. Problem Solving

    Points A(2, 8) and B(8, 2) are given.

    • (a) Find the gradient of AB and the gradient of the perpendicular bisector of AB.
    • (b) Find the midpoint of AB and write the equation of the perpendicular bisector.
    • (c) Point C lies on the perpendicular bisector of AB. If C has x-coordinate 3, find C.
    • (d) Verify that CA = CB (C is equidistant from A and B).