Equations with two unknowns

Mangoes cost ₹60 a kg and bananas ₹50 a kg, and the bill is ₹280. With x kg of mangoes and y kg of bananas: 60x + 50y = 280. Both sides measure the same thing: the cost in rupees.

Is it linear in two variables? Count the variables (two: x and y), then check the powers: each appears only to the first power, with no x², no xy and no x in a denominator.

Standard form: ax + by + c = 0, with a and b not both zero. a and b are the coefficients; c is the constant.

Move everything to the left, signs and all. 4 = 1.5x – 1.3y becomes 1.5x – 1.3y – 4 = 0, so a = 1.5, b = –1.3, c = –4.

Fractions are allowed: x/4 + 3y/5 – 1/2 = 0. Multiplying by 20 gives the neater 5x + 12y – 10 = 0.

x = 8 is also a linear equation in two variables: x + 0·y – 8 = 0. Here the coefficient of y is zero; y is still a variable.

Check your translation with numbers. 'A notebook costs twice a pen' is t = 2p: a ₹10 pen gives a ₹20 notebook, and 20 = 2 × 10. Writing p = 2t would make the pen dearer.