Same Difference, Every Time

A school canteen sells juice boxes in trays of 6 and biscuit packets in trays of 4. If Meera buys p trays of juice and q trays of biscuits, and the canteen throws in 2 extra juice boxes for free, the total number of items she walks away with is 6p + 4q + 2. In this expression, 6p, 4q and 2 are called terms: the parts added together to make the whole expression. The letters p and q are variables, standing in for a quantity that can change from one trip to the canteen to the next. The numbers 6 and 4, multiplying the variables, are called coefficients, and the lone number 2, with no letter attached, is the constant term. Every algebraic expression, however complicated it looks, is built from exactly these ingredients: terms, added or subtracted, each made of a coefficient times a variable, or just a number on its own for the constant.

Not every expression uses more than one letter. The expression 7z^3 - 2z^2 + z - 5 involves only the single variable z, and expressions like this, built from one variable and its powers, are called univariate polynomials, or simply polynomials. The highest power of the variable appearing in a polynomial is called its degree, and the degree gives the polynomial its name. In 7z^3 - 2z^2 + z - 5, the highest power of z is 3, so this is a polynomial of degree 3, called a cubic polynomial. A polynomial of degree 2, such as z^2 + z, is called quadratic. A polynomial of degree 1, such as z - 5, is called linear. Even a single number, like 5, counts as a polynomial: since 5 can be written as 5z^0 (anything to the power 0 is 1), it has degree 0, and is called a constant polynomial. This chapter is about the degree-1 case: linear polynomials, and the patterns they describe.

A neighbourhood library charges a one-time joining fee of ₹150, plus ₹30 for every book borrowed. If m is the number of books borrowed, the total amount paid is 150 + 30m, a linear polynomial in m. Making a small table for m = 1, 2, 3, 4 gives amounts of ₹180, ₹210, ₹240 and ₹270. Notice that the amount jumps by exactly ₹30 every time one more book is borrowed, no matter which stage of the pattern is checked: the difference between consecutive amounts is always the same constant value. This is the defining feature of a linear pattern: a sequence of numbers in which the difference between any two consecutive terms stays fixed. It is exactly this constant-difference property that sets a linear polynomial's pattern apart from every other kind.

Linear expressions are especially good at describing situations that grow or shrink by a fixed amount at every step. Consider a water tank that starts with 40 litres and is filled at a steady rate of 8 litres every minute: the volume after t minutes is V(t) = 40 + 8t. After 1, 2 and 3 minutes, the tank holds 48, 56 and 64 litres: the amount increases by a constant 8 litres each minute. This is linear growth. Now consider a candle that starts 24 cm tall and burns down at a steady 1.5 cm every hour: its height after t hours is h(t) = 24 - 1.5t. After 1, 2 and 3 hours, the candle stands at 22.5 cm, 21 cm and 19.5 cm: the height decreases by a constant 1.5 cm each hour. This is linear decay. Growth and decay are really the same idea wearing different signs: growth adds a fixed positive amount at every step, while decay subtracts a fixed amount, and both are perfectly described by a straight-line rule.

Sometimes the linear rule connecting two quantities isn't given directly, only a couple of clues are. A home-delivery service charges a fixed fee plus a fixed rate per kilometre, so the total charge y for a delivery of x kilometres follows y = ax + b. Suppose a delivery of 8 km costs ₹130, and a delivery of 14 km costs ₹190. Substituting both into y = ax + b gives two equations: 130 = 8a + b, and 190 = 14a + b. Subtracting the first equation from the second eliminates b entirely: 190 - 130 = 14a - 8a, so 60 = 6a, giving a = 10. Substituting back, b = 130 - 8(10) = 50. So the full rule is y = 10x + 50: a fixed fee of ₹50, plus ₹10 per kilometre. Two clues were all it took to pin down the entire linear relationship, because a straight line is completely determined by any two points on it.

Every linear relationship y = ax + b can be drawn as a straight line, and the two numbers a and b have very concrete meanings on the graph. The constant b is where the line crosses the y-axis (found by setting x = 0), so it is called the y-intercept. The number a controls the steepness and direction of the line, and is called the slope: a positive slope means the line rises from left to right, matching linear growth, while a negative slope means it falls from left to right, matching linear decay. To draw the line for y = 3x + 2, just two points are needed: at x = 0, y = 2, giving the point (0, 2); at x = 2, y = 8, giving the point (2, 8). Plotting these two points and drawing a straight line through them, extended in both directions, gives the complete graph, since a straight line is entirely determined once any two of its points are fixed.

Two delivery-style pricing plans might charge exactly the same rate per kilometre but have different fixed fees: say y = 10x + 50 for one company and y = 10x + 90 for another. Both lines have the same slope, 10, but different y-intercepts, 50 and 90. Plotted on the same graph, these two lines never meet: they run alongside each other at a constant vertical gap of ₹40, for every value of x. This is exactly what it means for two lines to be parallel: equal slopes, but different y-intercepts. If instead the y-intercepts were equal but the slopes differed, the two lines would share exactly one point, where they both cross the y-axis, and then separate, which is a completely different, intersecting relationship, not a parallel one. Recognising this pattern instantly tells you whether two linear rules will ever agree at some value of x, without drawing a single point.