Two Variables, One Line

An equation ax + by + c = 0 has infinitely many solutions, and they all lie on one straight line.

Key ideas

A linear equation in two variables is ax + by + c = 0 with a and b not both 0.
A solution is a pair (x, y) that makes the equation true; there are infinitely many.
To find solutions, choose a value of x and work out y.
Every solution is a point on the graph, which is a straight line; every point on the line is a solution.
x = k is a line parallel to the y-axis; y = k is parallel to the x-axis.
To check a point, substitute it: it lies on the line only if both sides are equal.

General form	ax + by + c = 0
Where the line meets the y-axis	put x = 0
Where the line meets the x-axis	put y = 0
Fare-type rule	y = fixed charge + rate × x

A taxi charges Rs 46 for the first km and Rs 16 for every km after that. Write an equation for the fare y for x km, and find the fare for 12 km.

Working: y = 46 + 16(x − 1) = 16x + 30; y = 16 × 12 + 30 = 222

Answer: y = 16x + 30; Rs 222