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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 · y − y₁ = m(x − x₁)substitute · put a number in place of a letterhorizontal line · y = constant, slope 0vertical line · 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 − y₁) ÷ (x − x₁) = m, so y − y₁ = m(x − x₁). Example: slope 3 through (2, 1): y − 1 = 3(x − 2), so y = 3x − 5. For a line through two points, first find m = (y₂ − y₁) ÷ (x₂ − x₁), then use either point. Example: (1, 2) and (3, 8): m = 6/2 = 3, so y − 2 = 3(x − 1), which gives y = 3x − 1. Check with the other point: 3 × 3 − 1 = 8. A horizontal line through (3, 4) is y = 4 and a vertical line through (3, 4) is x = 3.
Worked examples
Line through (−2, 5) and (4, −1). m = (−1 − 5)/(4 + 2) = −1. y − 5 = −1(x + 2), so y = −x + 3. Check: −4 + 3 = −1 ✓.
Slope 1/2 through (4, 3): y − 3 = (1/2)(x − 4), so y = x/2 + 1. Check at x = 4: 2 + 1 = 3 ✓.
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 = −2 and h = −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).
y − 5 = 2(x − 1), y = 2x + 3.
2
Write the line through (0, 1) and (2, 7).
m = 3, y = 3x + 1.
3
Write the line through (1, 4) and (5, 4).
Same y: horizontal, y = 4.
4
Write the line through (3, −1) and (3, 6).
Same x: vertical, x = 3.
5
Does the point (4, 7) lie on y = 2x − 1?
2 × 4 − 1 = 7: yes.