☰ Year plan · Mathematics

The equation of a straight line

Week 13 · The equation of a straight line: slope and y-intercept, point and slope, two points, standard form and applications · 3 lessons
3

The standard form Ax + By = C, parallel and perpendicular lines, and linear stories

Mathematics Prep 3 Term 1, Unit 3, Lesson 7
GoalUse Ax + By = C: find slope and intercepts, parallel and perpendicular lines, and solve linear stories.
New words
standard form · Ax + By = C with A, B, C whole numbers and A ≥ 0x-intercept · where the line cuts the x-axis, put y = 0clear fractions · multiply the equation by a number to remove fractionsconstant · a number that does not change
Explanation

A line can be written as Ax + By = C. From it: the slope is −A/B, the y-intercept is C/B and the x-intercept is C/A. Example: 2x + 3y = 12. The slope is −2/3. Put x = 0: 3y = 12, y = 4. Put y = 0: 2x = 12, x = 6. Joining (6, 0) and (0, 4) draws the line. To change y = mx + b to standard form, clear fractions and move the terms: y = −(2/3)x + 4 gives 3y = −2x + 12, so 2x + 3y = 12. Lines with the same A : B are parallel: 2x + 3y = 5 is parallel to 2x + 3y = 12 (and passes through (1, 1)). A perpendicular line has slope 3/2, for example 3x − 2y = 0.

xy−112345678−112345602x + 3y = 12(6, 0)(0, 4)
Worked examples
Write y = 3x − 1 in standard form: 3x − y = 1. Here A = 3, B = −1, C = 1, so the slope is −3/(−1) = 3, the y-intercept is 1/(−1) = −1 and the x-intercept is 1/3.
Line through (1, 1) parallel to 2x + 3y = 12: use the same left side, 2x + 3y = 2 + 3 = 5.
Tickets: adults pay 20 LE, children 10 LE and the family pays 100 LE: 20x + 10y = 100, that is 2x + y = 10. With 3 adults, y = 10 − 6 = 4 children. The slope is −2.
Class activity

Intercept hunt. Draw 3x + 4y = 12 using only its two intercepts, then check that both (4, 0) and (0, 3) lie on it.

Practice
1
Find the intercepts and slope of 3x + 4y = 12.
2
Write y = x/2 + 3 in standard form.
3
Line through (0, 0) perpendicular to 2x + 3y = 12.
4
A shop sells pens at 4 LE and notebooks at 10 LE. Write the number of pens x and notebooks y for 60 LE exactly, and find y if x = 5.
5
Are 2x + 3y = 12 and 3x − 2y = 5 perpendicular?