Chapter 7.4 — Exercise 7.4 — Slope of a Line

Two Ways to See the Same Number

Along any straight line, the ratio of vertical change to horizontal change between two points always comes out the same, no matter which two points on the line you pick — on the line through (1,2), (2,3), (3,4), (4,5), that ratio is always exactly 1. This constant ratio is the line's slope.

The same number has a second identity too: if a line makes angle θ with the positive x-axis, dropping perpendiculars from two points on the line onto a shared reference creates a right triangle whose opposite-over-adjacent ratio is tanθ — and that ratio turns out to be identical to (y₂−y₁)/(x₂−x₁). So slope isn't just a coordinate ratio; it's also literally the tangent of the angle the line makes with the x-axis.

The sign of m carries its own meaning before any arithmetic is even finished: a positive slope means the line climbs as you move left to right, a negative slope means it falls, a slope of exactly zero means the line is perfectly flat, and a line that runs straight up and down has no defined slope at all, since its "run" would be zero and division by zero simply isn't allowed. Predicting the sign of an answer before computing it — just by glancing at whether the line seems to rise or fall between the two given points — is a quick way to catch an arithmetic slip immediately, rather than only noticing something went wrong several steps later.

Eight Slopes, One Formula

A Point With a Root, Handled Carefully

The second row deserves its own look, since it's the only pair built from an irrational coordinate: (0,0) and (√3, 3). Substituting directly gives 3/√3 — a fraction with a surd sitting in the denominator, which is usually cleared by multiplying top and bottom by √3:

A slope of exactly √3 is worth recognising on sight, since it's also tan60° — meaning this particular line rises at a 60° angle from the x-axis, a fact the plain coordinate ratio doesn't announce on its own but the angle identity behind the formula makes visible. Rationalising the denominator here isn't just a cosmetic tidy-up either — leaving the answer as 3/√3 would still be numerically correct, but √3 is the form that makes the tan60° connection immediately recognisable, which is exactly why clearing a surd from the denominator is worth doing as a matter of habit whenever a slope calculation produces one.

A Slope of Exactly Zero

The sixth row, (3,−2) and (−6,−2), has identical y-coordinates at both points — the line joining them never rises or falls at all, giving a numerator of exactly 0 and therefore a slope of 0. A horizontal line is the one case where the numerator vanishes cleanly rather than the calculation needing any further simplification; a vertical line, by contrast, would make the denominator vanish instead, which is exactly why a vertical line's slope is considered undefined rather than zero. Spotting equal y-coordinates (or equal x-coordinates) before reaching for the full formula is worth doing as a first check on any pair of points, since it settles the answer immediately rather than requiring a subtraction that was always going to come out as zero anyway.

The third and fourth rows are worth grouping together too, since both are built from letters rather than numbers, and both simplify by cancelling a shared factor rather than by any numeric arithmetic at all. In (2a,3b) and (a,−b), the numerator −b−3b=−4b and the denominator a−2a=−a share no common numeric factor, but the negative signs on both top and bottom cancel cleanly to leave 4b/a. In (a,0) and (0,b), by contrast, only one sign flips, leaving the negative in the final answer: −b/a. Working through the sign of each term separately, rather than cancelling the two negatives by eye, is the safer habit whenever a slope calculation is built entirely from variables instead of specific numbers.

Fractions and Decimals, Same Formula Throughout

Rows five and seven are worth a second look purely for their arithmetic: decimal coordinates like (−1.4,−3.7) substitute into the slope formula exactly as whole numbers do, and mixed-number coordinates like (−3½,3) are easiest to handle by converting fully to improper fractions first, rather than trying to subtract a whole number and a half separately. Neither decimals nor fractions change the formula itself — they only change how carefully the subtraction step needs to be done. The eighth row, (0,4) and (4,0), is worth a final glance too: it's the mirror image of a very familiar shape, a line crossing both axes at equal distance from the origin, and its slope of exactly −1 reflects that symmetry directly — swap the roles of x and y in any point on this particular line and the point still lies on it.

Slope also connects directly back to a question this course already answered a different way. Chapter 4 classified a pair of linear equations as intersecting, parallel, or coincident purely by comparing the ratios of their coefficients — a1/a2 versus b1/b2. Slope gives the exact same classification a more geometric face: two lines are parallel (or the same line) precisely when they share the same slope, and they cross at exactly one point whenever their slopes differ. Nothing about the underlying mathematics has changed between the two chapters — comparing coefficient ratios and comparing slopes are simply two different ways of asking whether two lines point in the same direction.

The Chapter, From Distance to Direction

Chapter 7 built up four genuinely different ways to read meaning out of a pair of coordinates: Exercise 7.1 measured the straight-line gap between two points, Exercise 7.2 located a point dividing that gap in a stated ratio, Exercise 7.3 measured the region enclosed by three such points, and this exercise measured the steepness of the line joining any two of them. All four formulas share the same starting ingredients — two labelled points, x₁,y₁,x₂,y₂ — substituted into a different arrangement each time, which is worth remembering the next time any one of the four formulas seems to blur together with another. The next chapter, Similar Triangles, moves away from coordinates entirely, returning to pure geometric proportion and the relationships between triangles that share the same shape at different sizes.