Chapter Overview
Chapter 13 links a linear equation in two variables to all the points on a straight line. The chapter uses tables, coordinate graphs, slope, intercepts and pairs of equations to model everyday situations.
These worked solutions show the mathematical reasoning behind selected exercise questions and end-of-chapter problems.
Textbook-check note: Use the exact question numbers and diagrams in your copy of Ganita Manjari. Where data are read from a graph, answers are marked as estimates. This page does not reproduce every printed exercise verbatim.
Important Concepts
| Concept | Explanation |
|---|---|
| Linear equation in two variables | An equation of the form ax + by + c = 0, where a and b are not both zero. |
| Solution | An ordered pair (x,y) that makes the equation true. |
| Slope | For two different points, slope = (change in y)/(change in x). |
| Y-intercept | The point where the line crosses the y-axis. |
| System of equations | Two linear equations may have one, no, or infinitely many common solutions. |
Exercise Solutions — Chapter 13
Exercise Set 13.1 — Standard form and modelling
Question 1
For a = 3, b = 0 and c = −1/5, ax + by + c = 0 becomes 3x − 1/5 = 0.
Question 2 — Write in standard form
(i) y − 15 = √2x ⇒ √2x − y + 15 = 0.
(ii) 3y − 2x = 0 ⇒ −2x + 3y = 0.
(iii) 5x = 3y ⇒ 5x − 3y = 0.
(iv) x = 8 ⇒ x + 0y − 8 = 0.
(v) 3y = 1 ⇒ 0x + 3y − 1 = 0.
Question 3 — Representing situations
(i) If a notebook costs twice as much as a pen, let the prices be t and p. Then t = 2p.
(ii) If two batsmen together score 176 runs and their scores are x and y, then x + y = 176.
Exercise Set 13.2 — Solutions and graphs
Question 1 — Check (4,3)
Substitute into 5x − 6y = 2: 5(4) − 6(3) = 20 − 18 = 2. Therefore (4,3) is a solution.
Question 2 — Find two solutions each
(i) 7x − 3y = 21. If x = 0, y = −7, giving (0,−7). If y = 0, x = 3, giving (3,0).
(ii) 2x + 3y = 5. Taking x = 1 gives y = 1, hence (1,1). Taking x = −1 gives 3y = 7, hence (−1,7/3).
End-of-Chapter Exercises
Question 1 — Graph y = 3x
(i) Using (0,0) and (2,6), slope = (6−0)/(2−0) = 3.
(ii) The y-intercept is 0; the line passes through the origin, so the model starts at 0 cm.
(iii) (a) At x = 5, y = 3×5 = 15 cm. (b) 21 = 3x ⇒ x = 7 minutes.
(iv) At x = 4, y should be 12, not 10, so (4,10) is not on the line.
(v) When x = 3, y = 9; intersection is (3,9).
Question 2 — Convert Celsius and Fahrenheit
Use F = (9/5)C + 32.
(i) Plot points (0,32), (100,212) and (−40,−40), then join them with a straight line.
(ii) At 30°C: F = (9/5)×30 + 32 = 86°F.
(iii) 95 = (9/5)C + 32 ⇒ C = 35°C.
(iv) 0°C = 32°F. For 0°F, C = −160/9 ≈ −17.8°C.
(v) Put F = C: C = (9/5)C + 32 ⇒ C = −40°.
Question 3 — Solve graphically
For 2x + y = 6, points (0,6) and (3,0) lie on the line. For 2x − y = 2, points (0,−2) and (1,0) lie on the line. Substituting x = 2, y = 2 satisfies both equations, so the lines intersect at (2,2).
Question 8 — Pencil and pen prices
Let pencil cost x and pen cost y. Then 5x + 7y = 50 and 7x + 5y = 46. Adding gives x + y = 8. Subtracting gives y − x = 2. Solving: y = 5 and x = 3. Pencil ₹3; pen ₹5.
Question 9 — Height of a stool
Let stool height s and cat height c. Cat sitting on stool: s + c = 85. Cat on floor, with its head 25 cm below the stool top: s − c = 25. Adding gives 2s = 110, so s = 55 cm and c = 30 cm.
One Minute Revision
- A linear equation in two variables usually has infinitely many ordered-pair solutions.
- Two distinct points determine a straight line; a third point can check the graph.
- Slope = (y₂ − y₁)/(x₂ − x₁).
- A pair of equations is solved at the intersection of their graphs.
- Parallel distinct lines have no solution; coincident lines have infinitely many.
Frequently Asked Questions
1. How can we check an ordered pair?
Substitute both coordinates into the equation and verify the equality.
2. What is the slope of y = 3x?
Its slope is 3, because y increases by 3 units whenever x increases by 1.
3. At what temperature are Celsius and Fahrenheit numerically equal?
At −40°, since F = (9/5)C + 32 gives C = −40 when F = C.