Exercise 7.4 from Chapter 7, Coordinate Geometry, of Class 10 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) introduces the slope (also called the gradient) of a line — a single number that captures exactly how steeply a line rises or falls as you move along it. Once you know two points on a line, slope can be calculated in seconds, and it becomes one of the most frequently reused ideas in the rest of coordinate geometry.
This exercise contains a single question with eight parts, covering plain numeric points, points written with algebraic letters, decimals, mixed numbers, and a perfectly horizontal line — giving you a complete tour of every situation the slope formula can throw at you.
💡 Foundation fact: The slope of the line through (x₁, y₁) and (x₂, y₂) is
m = (y₂ − y₁) / (x₂ − x₁) — the change in y divided by the change in x. This single ratio also equals tan θ, where θ is the angle the line makes with the positive direction of the x-axis.
Reading the Sign of a Slope
Before diving into the calculations, it's worth building a quick mental picture of what each type of slope value actually looks like. The numeric value matters, but its sign (and whether it exists at all) tells you the line's basic shape at a glance.
Slope formula:m = (y₂ − y₁) / (x₂ − x₁)
Always subtract the coordinates in the same order in both the numerator and the denominator. Swapping which point you call "1" and which you call "2" flips the sign of both the top and the bottom — so the final ratio (the slope) never actually changes.
Positive slope (rises left → right)
Negative slope (falls left → right)
Zero slope (horizontal line)
Undefined slope (vertical line)
📌 Zero vs. undefined — don't mix them up: A horizontal line has slope exactly 0 (y never changes, so the numerator is 0). A vertical line has no slope at all — x never changes, so the denominator is 0, and division by zero is simply not defined. These are opposite situations, not the same thing.
Question 1 — Find the Slope of the Line Through the Given Points
Every part below follows the same two-line routine: label the two points, substitute into m = (y₂ − y₁) / (x₂ − x₁), and simplify. Watch the parts with negative numbers and fractions especially closely.
m = (3 − 0) / (3 − 0) = 3 / 3 = 1(A slope of exactly 1 means the line makes a 45° angle with the x-axis.)
Question 1(iii)
Points: (2a, 3b) and (a, −b)
m = (−b − 3b) / (a − 2a) = (−4b) / (−a) = 4b / a
📌 Algebraic points work exactly the same way — there's just no single numeric diagram to draw, since the actual steepness depends on whatever values a and b take. The formula and the bracket-handling are identical to the numeric parts; only the final simplification involves letters instead of numbers.
Question 1(iv)
Points: (a, 0) and (0, b)
m = (b − 0) / (0 − a) = −b / a
📌 A useful pattern to remember: (a, 0) and (0, b) are the x-intercept and y-intercept of a line. Whenever a line crosses the axes at these two points, its slope is always −(y-intercept)/(x-intercept) — a shortcut worth recognising in later "intercept form" problems.
⚠️ Most common error: Subtracting in a different order top and bottom — for example, writing (y₂ − y₁) / (x₁ − x₂) instead of (y₂ − y₁) / (x₂ − x₁). This silently flips the sign of your final answer. Always pick which point is "1" and which is "2" once, then stick with that choice for both the numerator and the denominator.
Common Mistakes to Avoid
Inconsistent subtraction order: The single biggest source of sign errors in this exercise. Decide once which point is (x₁, y₁) and which is (x₂, y₂), then use that same labelling in both the top and bottom of the fraction.
Confusing "0 slope" with "no slope": A slope of 0 means a horizontal line (Question 1(vi)); an undefined ("no") slope means a vertical line, where the formula's denominator becomes zero. They describe opposite situations.
Sign mistakes with double negatives: Parts like (v) involve subtracting negative decimals — e.g. −2.4 − (−1.4). Rewrite as addition of the opposite (−2.4 + 1.4) before simplifying, rather than trying to track two minus signs mentally.
Mishandling mixed numbers: In part (vii), convert mixed numbers like 3½ and 2½ into improper fractions (7/2 and 5/2) before subtracting — this avoids errors when borrowing across the whole-number and fractional parts.
Treating literal (algebraic) points as a different kind of problem: Parts (iii) and (iv) use the exact same formula as the numeric parts — the only extra step is simplifying an algebraic fraction at the end instead of a numeric one.
⛔ High-risk exam trap: When a question gives points with mixed numbers or mixed positive/negative decimals, examiners are specifically testing whether you can avoid sign errors under time pressure. Slow down by one extra step on these parts rather than rushing — the formula itself never changes.
Quick Reference — All Answers at a Glance
Part
Points
Slope (m)
(i)
(4, −8), (5, −2)
6
(ii)
(0, 0), (3, 3)
1
(iii)
(2a, 3b), (a, −b)
4b / a
(iv)
(a, 0), (0, b)
−b / a
(v)
(−1.4, −3.7), (−2.4, 1.3)
−5
(vi)
(3, −2), (−6, −2)
0
(vii)
(−3½, 3), (−7, 2½)
1/7
(viii)
(0, 4), (4, 0)
−1
Frequently Asked Questions
What does a negative slope mean?
A negative slope means the line falls as you move from left to right — y decreases while x increases. Parts (v) and (viii) of this exercise are both examples of falling, negative-slope lines.
What is an "undefined" slope, and how is it different from a slope of zero?
An undefined slope occurs for a vertical line, where the x-coordinate never changes — this makes the denominator of the slope formula zero, and division by zero has no defined value. A slope of zero, by contrast, occurs for a horizontal line, where the y-coordinate never changes, making the numerator (not the denominator) zero.
Does it matter which point I call (x₁, y₁) and which I call (x₂, y₂)?
No. Swapping the two points flips the sign of both the numerator and the denominator at the same time, so the overall ratio — the slope — stays exactly the same. What matters is staying consistent: use the same order in the top and the bottom of the fraction.
How is slope related to the angle a line makes with the x-axis?
Slope equals tan θ, where θ is the angle the line makes with the positive direction of the x-axis, measured anticlockwise. A 45° line has slope tan 45° = 1, which is exactly what Question 1(ii) demonstrates.
Can a slope be a fraction, or does it have to be a whole number?
Slope can be any rational number — a whole number, a fraction, or a negative value — depending entirely on the two points chosen. Question 1(vii) in this exercise gives a slope of 1/7, showing that gentle, slowly-rising lines have small fractional slopes.
What This Exercise Prepares You For
Slope is one of the most-reused ideas in coordinate geometry: it's the standard way to check whether two lines are parallel (equal slopes) or perpendicular (the product of their slopes equals −1), and it underpins how the equation of a line is written and interpreted in later topics. The sign-handling and careful substitution practised across all eight parts here — especially with negative numbers, decimals, and mixed numbers — is exactly the same care needed when working with the linear equations met earlier in the course.
If you've already studied pairs of linear equations in two variables, you'll notice that slope is really just a restatement of the same idea using a different formula — both describe how steeply a line tilts, just arrived at from different starting information (an equation versus two points).
📐 Board Exam Tip (Telangana & AP): Slope questions are often combined with a follow-up like "show that the line through these points is parallel to another line" or "find if three points are collinear using slope" (two pairs of points give the same slope only when all three points lie on one line). Practising the plain calculation here, as in all eight parts of Question 1, makes those combined questions much faster to answer.
