Solving by elimination

The idea: choose the letter you want to get rid of. Make its numbers the same in both equations, then subtract (or add) to wipe it out.

Example: the incomes of two people are in the ratio 9 : 7 and their spending is in the ratio 4 : 3. Each saves ₹2000 a month. So the incomes are 9x and 7x, the spending is 4y and 3y, and income – spending = savings:

9x – 4y = 2000 and 7x – 3y = 2000

• Step 1: to make the y's match, multiply the first by 3 and the second by 4. Multiply every term, both sides: 27x – 12y = 6000 and 28x – 12y = 8000.
• Step 2: subtract. When you subtract a whole equation, every sign in it flips: x = 2000.
• Step 3: put it back: 9(2000) – 4y = 2000, so y = 4000.
• Answer: the incomes are ₹18,000 and ₹14,000. Check: 16,000 : 12,000 = 4 : 3 ✓

Add or subtract? If the matching terms have the same sign, subtract. If they have opposite signs (+4y and –4y), add.

If everything vanishes: 2x + 3y = 8 and 4x + 6y = 7. Double the first: 4x + 6y = 16. Subtract: 0 = 9, which is false, so no solution.

Two-digit numbers. If the tens digit is x and the units digit is y, the number is 10x + y, and with the digits reversed it is 10y + x. A number plus its reverse is 66, so 11x + 11y = 66 and x + y = 6.

'The digits differ by 2' gives two cases: x – y = 2 or y – x = 2. That leads to two answers, 42 and 24. Writing only one is an incomplete answer.

A tip for big numbers: if the equations are like 131x + 217y = 913 and 217x + 131y = 827, add them and subtract them. You get x + y = 5 and y – x = 1, so x = 2 and y = 3.