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Cambridge IGCSE0607

Introduction to algebra

International Mathematics 0607 Chapter Notes

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Introduction to algebra
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Introduction to algebra notes

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In this chapter

  1. 1Letters stand for numbers
  2. 2Substituting into a formula
  3. 3Negative numbers and order of working

1. Letters stand for numbers

A letter can stand for any number. It is a generalised number. This lets us write a rule once so that it works for every case.

Write a product with no sign: 6x means 6 × x. A formula links one letter to others. A regular hexagon with side x has six equal sides, so its perimeter is six lots of x.

P = 6x

Worked example 14 marks

(a) A regular hexagon has side length x.

(i) Find a formula, in terms of x, for the perimeter, P, of a regular hexagon.

(ii) The perimeter of a regular hexagon is 72 cm. Use your formula in (a)(i) to work out the side length of the hexagon.

  1. 1

    a regular hexagon has 6 equal sides, each of length x [1]

  2. 2

    P = 6x [1]

  3. 3

    6x = 72 [1]

  4. 4

    x = 12 cm [1]

2. Substituting into a formula

To substitute, replace each letter with its number and work out the value. Write the formula first, then put the numbers in brackets.

If the answer is given and a letter is unknown, substitute the known values and solve the equation that is left.

Examiner insight

“Candidates did well here with only a few ignoring the brackets and so worked out 2 × 3 + 4 = 10.”

Worked example 12 marks

H = 2W - 3X

Find the value of W when H = 4 and X = 2.

  1. 1

    substitute: 4 = 2W - 3 × 2, so 2W = 4 + 6 = 10 [1]

  2. 2

    W = 5 [1]

3. Negative numbers and order of working

When a letter is replaced by a negative number, put the number in brackets. Then follow the order of operations: brackets, powers, multiplication and division, then addition and subtraction.

A power belongs only to the letter it is written on. 3a² means 3 × a², not (3a)². Squaring a negative number gives a positive one: (-7)² = 49. Subtracting a negative adds: 5 - (-3) = 8.

Examiner insight

“Some did not use the correct order of operations, giving their working as –7 × 7 = –49.” “The common error was the manipulation of negative signs when finding 5 – (–3) or (–4) – (–2).”

Worked example 13 marks

Do not use a calculator.

Work out the value of 3a² - 2ab when a = -2 and b = 5.

  1. 1

    3a² = 3 × (-2)² = 3 × 4 = 12 [1]

  2. 2

    2ab = 2 × (-2) × 5 = -20 [1]

  3. 3

    12 - (-20) = 12 + 20 = 32 [1]

Recap

  • A letter stands for any number: a generalised number.
  • 6x means 6 × x.
  • To substitute, write the formula, then put each number in brackets.
  • Follow the order: brackets, powers, multiply or divide, add or subtract.
  • Squaring a negative number gives a positive number.
  • Subtracting a negative number adds.
  • If the answer is given, substitute the known values and solve.

End-of-chapter exercise

Test yourself on the whole chapter. Work through these before moving on.

  1. A pen costs p cents. Write down an expression for the cost, in cents, of 5 pens.1 mark
  2. Do not use a calculator. Work out 3x + 4 when x = 6.2 marks
  3. A taxi charges a fixed fee of 4 dollars plus 2 dollars for each kilometre. Write a formula for the total cost C dollars of a journey of k kilometres.2 marks
  4. Do not use a calculator. a = 4 and b = -3. Work out (a) a - b and (b) ab.2 marks
  5. Do not use a calculator. Work out 5 - 2x² when x = -3.2 marks
  6. The area of a trapezium is A = 1/2h(a + b). Work out A when h = 6, a = 5 and b = 9.2 marks
  7. s = ut + 1/2at². Work out s when u = 8, a = -2 and t = 3.3 marks
  8. Do not use a calculator. F = 9C/5 + 32. Find C when F = 50.3 marks
  9. Do not use a calculator. A cable hangs between two towers of a bridge. At a horizontal distance of x metres from the centre of the bridge, the cable is h metres above the road, where h = x²/8 - x/2 + 3. Points to the left of the centre have negative values of x. (a) Find h when x = -4. (b) Sana says the cable is at the same height at x = 4 as at x = -4. Use the formula to decide whether she is correct.5 marks
  10. On a windy day the air feels colder than it is. The feels-like temperature, F °C, is estimated by F = T - (v(10 - T))/20, where T °C is the air temperature and v km / h is the wind speed. (a) Find F when T = -6 and v = 15. (b) On another day T = -2 and F = -14. Find the wind speed.6 marks

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