0580 · 2.1Introduction to algebra

Variables, expressions and formulas

0:00Three notebooks costing two pounds each have a total cost of six pounds. But a shop needs a rule that works for any number of notebooks. Algebra lets us write that rule using a letter instead of choosing one particular number. By the end, you will understand what those letters represent and how to substitute values into expressions and formulas. You already need basic arithmetic, including negative numbers, fractions and the order of operations.

0:22Work inside brackets first, then calculate powers, before multiplication and division. Addition and subtraction come afterwards. Substitution keeps all of these rules. It simply turns algebra into ordinary arithmetic. Letters can be used to represent generalised numbers. This means one letter can stand for different possible numerical values, rather than one particular value chosen in advance. In our notebook rule, n represents the number bought. The expression 2n represents two multiplied by that number.

0:54A letter whose value can vary is called a variable. It is not necessarily a mysterious number that we must find. Sometimes its value is given, and sometimes it describes a quantity that can change. Within one calculation, however, every occurrence of the same letter represents the same value. The situation determines which values make sense. For counting whole notebooks, n must be zero or a positive whole number. In other calculations, a letter might represent a negative number, a decimal or a fraction.

1:21Letters do not automatically mean positive whole numbers, and different letters can sometimes have equal values. Algebra often leaves out the multiplication sign. 3x means 3 × x, not a number with two digits. Similarly, the letter a written next to the letter b means a × b. Meanwhile, x² means x multiplied by itself, which is different from 2 × x. An expression is a mathematical combination of numbers, letters and operations, such as 3x + 5.

1:51It does not state that two quantities are equal. A formula is an equation expressing a relationship between quantities. Our total cost rule becomes a formula when we write that cost equals 2n.

Substituting values into expressions

2:04Substitution means replacing each letter with its given numerical value. The operations stay exactly as written, because we are evaluating the existing rule rather than changing it. Write the substituted calculation before doing the arithmetic. Brackets around substituted negative numbers help keep their signs attached. For our first exam-style example, calculate 3x + 5 when x = 4. Replace x with four to obtain three multiplied by four plus five.

2:31Multiplication comes first, giving 12 + 5. Adding five gives the answer seventeen. Adding four and five first would give the wrong result because the expression contains no brackets around that addition. The multiplication applies to x alone. Also, 3x becomes 3 × 4, not thirty-four. Keeping the multiplication sign in your substituted working makes the meaning clear. For the second example, calculate a squared plus 2b, all divided by b minus a.

2:58We are given a equals negative three and b = 2. Substitute both values everywhere they occur, keeping negative three inside brackets. The fraction bar groups the entire numerator and the entire denominator. Squaring negative three gives positive nine, because a negative times a negative is positive. Next, doubling b gives four. The numerator therefore becomes 9 + 4, which equals thirteen. In the denominator, 2 − −3 gives five, so 13 ÷ 5 = 2.6.

3:30Two common errors become visible here. Squaring negative three does not mean making 3² negative, because the whole substituted number is being squared. Also, subtracting negative three means adding three, not subtracting three. Replace every occurrence of each letter before calculating so that none is accidentally left behind.

Using formulas and exam answers

3:49For our third example, a delivery service uses the formula C = 2 + 1.5n. Here, C is the cost in pounds, and n is the number of items. The two pounds is a fixed charge, while each item adds one pound fifty. Calculate the cost of delivering four items. Substituting four for n gives C equals two plus one point five multiplied by four. Calculate the multiplication first to obtain 2 + 6.

4:13Adding the fixed charge gives C = 8. The delivery cost is therefore eight pounds, including the charge that applies once to the whole order. In an examination, the command calculate asks you to work out the numerical value. Show the substitution, follow the order of operations, and include units when the formula represents a measured quantity. These methods must work without a calculator on Paper two. Paper four allows a calculator, but you must still enter the expression correctly.

4:40All three answers here are exact, so no rounding is needed. When a question requires a rounded answer, follow its stated accuracy. Otherwise, Cambridge generally requires answers that are not exact to three significant figures. Keep full intermediate values rather than rounding during the calculation. The central idea is that letters let one mathematical rule describe many possible numerical cases. Substitution selects a particular case by replacing those letters with given values.

5:06The operations remain unchanged, including powers, multiplication and the grouping shown by brackets or fraction bars. Once the replacement is correct, ordinary arithmetic completes the calculation.