☰ Year plan · Mathematics

The equation of a straight line

Week 13 Β· The equation of a straight line: slope and y-intercept, point and slope, two points, standard form and applications Β· 3 lessons
2

The line through a given point with a given slope, and through two points

Mathematics Prep 3 Term 1, Unit 3, Lesson 7
GoalWrite the equation of a line from a point and a slope, or from two points.
New words
point-slope form Β· y βˆ’ y₁ = m(x βˆ’ x₁)substitute Β· put a number in place of a letterhorizontal line Β· y = constant, slope 0vertical line Β· x = constant, no slope
Explanation

If a line has slope m and passes through the point (x₁, y₁), any other point (x, y) on it satisfies (y βˆ’ y₁) Γ· (x βˆ’ x₁) = m, so y βˆ’ y₁ = m(x βˆ’ x₁). Example: slope 3 through (2, 1): y βˆ’ 1 = 3(x βˆ’ 2), so y = 3x βˆ’ 5. For a line through two points, first find m = (yβ‚‚ βˆ’ y₁) Γ· (xβ‚‚ βˆ’ x₁), then use either point. Example: (1, 2) and (3, 8): m = 6/2 = 3, so y βˆ’ 2 = 3(x βˆ’ 1), which gives y = 3x βˆ’ 1. Check with the other point: 3 Γ— 3 βˆ’ 1 = 8. A horizontal line through (3, 4) is y = 4 and a vertical line through (3, 4) is x = 3.

xyβˆ’112345βˆ’6βˆ’4βˆ’224680y = 3x βˆ’ 5(2, 1)
Worked examples
Line through (βˆ’2, 5) and (4, βˆ’1). m = (βˆ’1 βˆ’ 5)/(4 + 2) = βˆ’1. y βˆ’ 5 = βˆ’1(x + 2), so y = βˆ’x + 3. Check: βˆ’4 + 3 = βˆ’1 βœ“.
Slope 1/2 through (4, 3): y βˆ’ 3 = (1/2)(x βˆ’ 4), so y = x/2 + 1. Check at x = 4: 2 + 1 = 3 βœ“.
A candle is 30 cm tall and burns 2 cm each hour. After 3 h it is 24 cm: points (0, 30) and (3, 24) give m = βˆ’2 and h = βˆ’2t + 30.
Class activity

Two-point line. Choose two points on squared paper, draw the line, read its y-intercept and slope, and write the equation. Check with a third point on the line.

Practice
1
Write the equation of the line with slope 2 through (1, 5).
2
Write the line through (0, 1) and (2, 7).
3
Write the line through (1, 4) and (5, 4).
4
Write the line through (3, βˆ’1) and (3, 6).
5
Does the point (4, 7) lie on y = 2x βˆ’ 1?