5.3 Slope, Intercepts & Determining the Equation of a Line

5.3 Slope, Intercepts & Determining the Equation of a Line

Skill 5 of Competency 2 spells out five distinct construction tasks: find the equation of a line that passes through two given points, through one given point, is perpendicular to a given line, is parallel to a given line, or has a given slope. Each has a preferred method.

Slope

Slope measures steepness as the ratio of vertical change to horizontal change:

m = (y₂ − y₁)/(x₂ − x₁) = rise/run = Δy/Δx

The subtraction order must be consistent: whichever point you call (x₁, y₁) in the numerator must play the same role in the denominator. Reversing both gives the same slope; reversing one gives the wrong sign.

Through (−2, 7) and (4, −5): m = (−5 − 7)/(4 − (−2)) = −12/6 = −2

Zero versus undefined is a guaranteed distractor pair. A horizontal line has zero rise, so m = 0/run = 0. A vertical line has zero run, so m = rise/0 is undefined. The line y = 5 is horizontal; the line x = 5 is vertical.

From a table, slope is the constant ratio Δy/Δx. If x increases by 3 while y increases by 12, m = 4. If the ratio is not constant, the relationship is not linear.

The three forms of a line

+---------------------------------------------------------------------------+
|  Slope-intercept   y = mx + b        m = slope, b = y-intercept           |
|  Point-slope       y - y1 = m(x - x1)   uses any point and the slope      |
|  Standard          Ax + By = C       A, B, C integers, A >= 0             |
+---------------------------------------------------------------------------+

In standard form, the slope is −A/B, the x-intercept is C/A, and the y-intercept is C/B. For 3x + 4y = 24: slope is −3/4, x-intercept is 8, y-intercept is 6.

The five construction tasks

1. Given slope and y-intercept. Write y = mx + b directly.

2. Given a point and a slope. Use point-slope.

Slope 3 through (2, −5): y + 5 = 3(x − 2) → y = 3x − 11

3. Given two points. Compute the slope, then use point-slope with either point.

Through (1, 9) and (5, −3): m = (−3 − 9)/(5 − 1) = −3. Then y − 9 = −3(x − 1) → y = −3x + 12. Using the other point gives y + 3 = −3(x − 5) → y = −3x + 12, the same line.

4. Parallel to a given line through a point. Parallel lines have equal slopes.

Parallel to y = (2/3)x − 1 through (6, 4): slope stays 2/3. Then y − 4 = (2/3)(x − 6) → y = (2/3)x.

5. Perpendicular to a given line through a point. Perpendicular slopes are opposite reciprocals, with m₁ · m₂ = −1.

Perpendicular to y = (2/3)x − 1 through (6, 4): the perpendicular slope is −3/2. Then y − 4 = (−3/2)(x − 6) → y = −(3/2)x + 13.

Take the reciprocal and change the sign. The opposite reciprocal of 4 is −1/4; of −5/2 is 2/5; of −1 is 1. The special case: a horizontal line and a vertical line are perpendicular, even though the product rule fails because one slope is undefined.

Intercepts and their meaning

The y-intercept is the point where x = 0; the x-intercept (or zero) is where y = 0.

For 5x − 2y = 20: setting x = 0 gives y = −10, so the y-intercept is (0, −10). Setting y = 0 gives x = 4, so the x-intercept is (4, 0).

In context these carry distinct meanings, and interpretation items are common:

A pool drains according to V = 4,800 − 160t, where V is gallons and t is hours. Slope −160: the pool loses 160 gallons per hour. y-intercept 4,800: the pool held 4,800 gallons at t = 0. x-intercept 30: the pool is empty after 30 hours.

Always state slope with units — "160 gallons per hour," not just "−160." Items frequently ask which statement correctly interprets the slope, and the wrong options reverse the units or confuse slope with intercept.

Reading slope from a graph

Count rise over run between two lattice points — points where the line crosses grid intersections exactly. Estimating between grid lines introduces error that changes the answer.

A useful check: a slope of 2 means the line rises 2 units for every 1 unit right, so it is fairly steep. A slope of 1/2 rises 1 unit for every 2 right, so it is shallow. Candidates who invert the ratio get 1/2 where 2 belongs, and both values appear as choices.

Proportional versus merely linear

A line through the origin has b = 0 and represents a proportional relationship y = mx, where m is the constant of proportionality. Any line with a nonzero y-intercept is linear but not proportional: doubling x does not double y. In the pool example, the relationship is linear but not proportional, since the pool starts with 4,800 gallons already in it.