Income–expense with equal savings: Two persons have weekly incomes in the ratio 7 : 3 and weekly expenses in the ratio 5 : 2. Each saves ₹ 300 per week. Find the weekly income of the first person by solving the pair of linear equations.

Introduction / Context:
When incomes and expenses are given in separate ratios and savings (income minus expense) are fixed, represent each with scaled variables and solve the resulting simultaneous equations. This is a classic linear system application in ratio problems.

Given Data / Assumptions:

• Incomes: 7x and 3x.
• Expenses: 5y and 2y.
• Savings for each = ₹ 300.

Concept / Approach:
Write equations for each person: 7x − 5y = 300 and 3x − 2y = 300. Solve the system for x and y, then compute the first person’s income 7x.

Step-by-Step Solution:

Verification / Alternative check:
From (2), 3*900 − 2y = 300 ⇒ y = 1200. Expenses: 5y = 6000 and 2y = 2400. Savings: 6300 − 6000 = 300 and 2700 − 2400 = 300, consistent.

Why Other Options Are Wrong:
₹ 5400, ₹ 4500, ₹ 7500, ₹ 7000 do not satisfy both savings equations simultaneously.

Common Pitfalls:
Treating the same scaling variable for both income and expense ratios or incorrectly eliminating variables when solving the system.

Final Answer:
₹ 6300