Ch 3: Linear Equations (Old Syllabus Topics)
Covers Cross-Multiplication Method and complex word problems (Upstream/Downstream, Work/Time) from Old Exercises 3.5 & 3.6.
Exercise 3.5 (Cross-Multiplication Method)
Q1. Solve the following pair of linear equations by cross-multiplication method:
2x + y = 5
3x + 2y = 8
Solution:
First, write in standard form (ax + by + c = 0):
2x + y - 5 = 0
3x + 2y - 8 = 0
Using Cross-Multiplication formula:
x / (b1c2 - b2c1) = y / (c1a2 - c2a1) = 1 / (a1b2 - a2b1)
x / [(1)(-8) - (2)(-5)] = y / [(-5)(3) - (-8)(2)] = 1 / [(2)(2) - (3)(1)]
x / (-8 + 10) = y / (-15 + 16) = 1 / (4 - 3)
x / 2 = y / 1 = 1 / 1
Therefore, x = 2 and y = 1.
Exercise 3.6 (Equations Reducible to Linear Form)
⚠️ Board Tip: These 4-mark questions involve substituting 1/x = u and 1/y = v.
Q1. Solve:
1/(2x) + 1/(3y) = 2
1/(3x) + 1/(2y) = 13/6
Solution:
Let 1/x = p and 1/y = q.
Equations become:
p/2 + q/3 = 2 ⇒ 3p + 2q = 12 (Eq 1)
p/3 + q/2 = 13/6 ⇒ 2p + 3q = 13 (Eq 2)
Solving these linearly (multiply Eq1 by 2 and Eq2 by 3):
6p + 4q = 24
6p + 9q = 39
Subtracting: -5q = -15 ⇒ q = 3.
Put q=3 in Eq 1: 3p + 6 = 12 ⇒ p = 2.
Since 1/x = p ⇒ 1/x = 2 ⇒ x = 1/2.
Since 1/y = q ⇒ 1/y = 3 ⇒ y = 1/3.
Q2 (ii). 2 women and 5 men can together finish an embroidery work in 4 days, while 3 women and 6 men can finish it in 3 days. Find the time taken by 1 woman alone and 1 man alone.
Solution:
Let 1 woman finish the work in x days, and 1 man in y days.
Work done by 1 woman in 1 day = 1/x.
Work done by 1 man in 1 day = 1/y.
Condition 1: 2/x + 5/y = 1/4
Condition 2: 3/x + 6/y = 1/3
Let 1/x = u and 1/y = v.
2u + 5v = 1/4 ⇒ 8u + 20v = 1 (Eq 1)
3u + 6v = 1/3 ⇒ 9u + 18v = 1 (Eq 2)
Solving these gives: u = 1/18 and v = 1/36.
Since 1/x = 1/18 ⇒ x = 18 days (Time taken by 1 woman).
Since 1/y = 1/36 ⇒ y = 36 days (Time taken by 1 man).