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Cambridge IGCSE0580

Introduction to algebra

Mathematics 0580 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 an expression
  3. 3Substituting into a formula

1. Letters stand for numbers

A letter can stand for any number. It is a generalised number.

So a letter lets you write a rule that works whatever the number is. If Bella is b years old, then Alan, who is 5 years older, is b + 5 years old.

Two numbers or letters written side by side are multiplied. The cost of x bags at 65 cents each is 65x cents.

Worked example 12 marks

(a) Bella is b years old. Alan is 5 years older than Bella. Write an expression, in terms of b, for how old Alan is.

(b) Write an expression for the total cost, in cents, of x bags of sweets costing 65 cents each.

  1. 1

    (a) 5 years older means add 5: b + 5 [1]

  2. 2

    (b) cost of one bag × number of bags: 65 × x = 65x cents [1]

2. Substituting into an expression

To substitute, replace each letter with its value. Write the whole sum out first, then work it out.

Follow the order of operations: powers first, then multiplication, then addition. A power applies only to the number or letter next to it, so 4x² means 4 × x².

Take care with negative numbers. A negative times a negative is positive.

Examiner insight

“Showing a full substitution of the values into the expression first usually resulted in the correct answer.” “In questions of this nature requiring two or three calculations, candidates should show some working since it was clear that at least one of these calculations was usually correct.”

Worked example 12 marks

Do not use a calculator. Find the value of 4x² + 3y when x = 5 and y = 10.

  1. 1

    write the substitution: 4 × 5² + 3 × 10 [1]

  2. 2

    power first: 4 × 25 + 30 = 100 + 30 = 130 [1]

3. Substituting into a formula

A formula is a rule linking letters, or words, to a result. Put the given values in place of the letters and work out the result.

Check which letter takes which value. A number that is already in the formula stays as it is.

Multiply before you add. In the question below, the rate is multiplied by the days before the fixed charge is added.

Examiner insight

“The main challenge was to understand that 65 had to be substituted for C in the formula and not to replace the 9, which led to an answer of 45.” “After correctly substituting into the formula, it was common to see incorrect order of operations applied”

Worked example 12 marks

A car hire company uses this formula to work out the hire cost, in dollars.

cost = fixed charge + rate per day × number of days

Find the cost of hiring a car for 7 days when the fixed charge is 140 dollars and the rate per day is 30 dollars.

  1. 1

    substitute: cost = 140 + 30 × 7 [1]

  2. 2

    multiply first: 140 + 210 = 350 dollars [1]

Recap

  • A letter stands for any number: a generalised number.
  • Write 5 more than b as b + 5, and x bags at 65 cents as 65x cents.
  • Substitute: replace each letter with its value, writing the whole sum out first.
  • Powers first, then multiplication, then addition.
  • A negative times a negative is positive.
  • Show every calculation. A wrong answer with no working scores nothing.

End-of-chapter exercise

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

  1. Do not use a calculator. A pen costs p cents. Write an expression, in cents, for the cost of 6 pens and a ruler that costs 40 cents.2 marks
  2. Maya is m years old. Her brother is 4 years younger than Maya. Her father is three times as old as Maya. Write an expression, in terms of m, for the age of her brother and for the age of her father.2 marks
  3. Do not use a calculator. Find the value of 3x + 7 when x = -2.2 marks
  4. Do not use a calculator. Find the value of x² - 5x when x = -3.3 marks
  5. Use a calculator. The speed v is found with v = u + at. Find v when u = 12, a = 0.8 and t = 15.3 marks
  6. Do not use a calculator. A taxi charges a fixed 3.50 dollars plus 1.20 dollars for each kilometre. cost = fixed charge + rate per kilometre × number of kilometres Find the cost of a journey of 10 kilometres.3 marks
  7. Use a calculator. Find the value of (a² - b)/2a when a = -4 and b = 3.3 marks
  8. Do not use a calculator. Find the value of s = ut + 1/2at² when u = 8, a = -10 and t = 2.4 marks
  9. Do not use a calculator. The formula F = 9C/5 + 32 changes a temperature of C degrees Celsius into F degrees Fahrenheit. At night the temperature in a desert is -15 °C. By midday the temperature is 50 °C higher. (a) Find the night temperature in degrees Fahrenheit. (b) Find the rise in temperature, in degrees Fahrenheit, from night to midday.5 marks
  10. Use a calculator. A plant pot has a square top of side a cm, a square base of side b cm and a height of h cm. Its volume, V cm³, is given by V = 1/3h(a² + ab + b²) (a) Find V when a = 7.5, b = 4.5 and h = 16. (b) Compost costs k cents for each cm³. Write an expression, in terms of k, for the cost, in cents, of the compost that fills 5 of these pots.5 marks

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