Coordinate Geometry • Topic 3 of 4

Slope & Lines

The slope of a line measures its steepness: m = (y₂−y₁)/(x₂−x₁), the rise over the run. A positive slope rises left to right, a negative slope falls, a horizontal line has slope 0, and a vertical line has an undefined slope (the run is zero). Once you have a slope and any one point you can write the line in several equivalent forms, and CAT rewards picking the one that matches the data. Slope–intercept form y = mx + c is best when you know the slope and where the line crosses the y-axis. Two-point form, (y−y₁) = m(x−x₁), is the default when two points are given. Intercept form x/a + y/b = 1 is the fast route whenever a question talks about where the line meets the axes — a is the x-intercept, b the y-intercept. From the general form ax + by + c = 0 the slope is −a/b. The two relationships you must reach for instantly: parallel lines have equal slopes (m₁ = m₂), and perpendicular lines have slopes whose product is −1 (m₁·m₂ = −1, so each is the negative reciprocal of the other). These two conditions answer most CAT line questions in a single line of algebra.

✅ Solved examples

1. Find the slope of the line through (1, 2) and (4, 11).
m = (11−2)/(4−1) = 9/3 = 3.
2. Find the equation of the line through (2, 3) with slope −2.
y − 3 = −2(x − 2) ⇒ y = −2x + 4 + 3 ⇒ y = −2x + 7, i.e. 2x + y = 7.
3. A line is perpendicular to y = (3/4)x + 1 and passes through (3, −2). Find its equation.
Perpendicular slope = −4/3. y + 2 = −(4/3)(x − 3) ⇒ y = −(4/3)x + 4 − 2 ⇒ y = −(4/3)x + 2, i.e. 4x + 3y = 6.
4. A line cuts intercepts of 4 on the x-axis and 6 on the y-axis. Find its equation and slope.
Intercept form: x/4 + y/6 = 1 ⇒ 3x + 2y = 12. Slope = −a/b = −3/2.

✏️ Practice — try these, take hints as needed

1. Find the slope of the line through (−2, 5) and (3, −5).
m = (y₂−y₁)/(x₂−x₁).
(−5 − 5)/(3 − (−2)).
−10/5.
−2
2. Write the equation of the line through (0, −4) with slope 3.
Use y = mx + c.
c is the y-intercept −4.
m = 3.
y = 3x − 4
3. Are the lines 2x + 3y = 7 and 4x + 6y = 5 parallel, perpendicular, or neither?
Slope of ax+by+c=0 is −a/b.
Both give −2/3.
Equal slopes ⇒ ?
Parallel
4. Find k so that the line through (1, k) and (4, 7) is perpendicular to a line of slope 3.
Required slope = −1/3.
(7 − k)/(4 − 1) = −1/3.
7 − k = −1.
k = 8
5. A line passes through (3, 4) and is parallel to 5x − 2y = 1. Find its equation.
Parallel ⇒ same slope 5/2.
y − 4 = (5/2)(x − 3).
Clear the fraction.
5x − 2y = 7

📝 Topic test — 8 questions

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