Substitution and elimination

The plan: turn two equations in two unknowns into one equation in one unknown.

Substitution. From x + 2y = 3, write x = 3 – 2y. Put it into 7x – 15y = 2: 21 – 29y = 2, so y = 19/29 and x = 49/29. Easiest when a variable already has coefficient 1.

Elimination. In 9x – 4y = 2000 and 7x – 3y = 2000, multiply by 3 and by 4 to make both y-terms –12y. Subtract: x = 2000, then y = 4000. Easiest when no coefficient is 1.

Always check, in both equations and in the story. Incomes of ₹18,000 and ₹14,000 are indeed in the ratio 9 : 7.

Swapped coefficients? Add and subtract. 9x + 7y = 107 and 7x + 9y = 101 (citrons and wood-apples, a puzzle from old Indian texts): adding gives x + y = 13, subtracting gives x – y = 3, so x = 8, y = 5.

Digit problems. A two-digit number with digits x and y is 10x + y; reversed, it is 10y + x.