Cambridge Lower Secondary CheckpointStage 7

Algebra (7Ae)

Mathematics Stage 7 Chapter Notes

What this chapter covers

Algebra - Expressions, equations and formulae
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Algebra (7Ae) notes

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In this chapter

  1. 1From Words to Algebra: Constructing Expressions
  2. 2Plugging in the Numbers: Substitution
  3. 3Tidying Up Algebra: Collecting Like Terms
  4. 4Multiplying Out: Expanding Single Brackets
  5. 5Finding the Unknown: Solving Simple Equations

1. From Words to Algebra: Constructing Expressions

Algebra uses letters, called variables, to represent unknown numbers. An expression is a mathematical sentence that combines numbers, variables, and operations like addition, subtraction, multiplication, and division. It does not have an equals sign. For example, if a book costs 'b' pounds and a pen costs 'p' pounds, the total cost of 2 books and 3 pens can be written as the expression 2b + 3p.

a × b is written as ab

k × 5 is written as 5k (number first)

a ÷ b is written as a/b

a × a is written as a²

Key term

Variable: A letter or symbol that represents an unknown or changing quantity in an algebraic expression.

Examiner insight

Examiners look for the ability to correctly translate verbal descriptions into symbolic algebra, paying close attention to the order of operations and terms.

Common pitfall

For a phrase like '5 less than x', a common mistake is writing 5 - x. The correct expression is x - 5, as you are taking 5 away from x.

Worked example 12 marks

A smartphone has 'p' gigabytes of storage. A memory card has 'm' gigabytes of storage. Write an expression for the total storage of 3 smartphones and 5 memory cards.

  1. 1

    Storage of 3 smartphones = 3 × p = 3p

  2. 2

    Storage of 5 memory cards = 5 × m = 5m

  3. 3

    Total storage is the sum of these two amounts.

  4. 4

    Expression: 3p + 5m

Worked example 22 marks

An adult cinema ticket costs £a and a child's ticket costs £c. Write an expression for the total cost for a family of 2 adults and 4 children.

  1. 1

    Cost for 2 adults = 2 × a = 2a

  2. 2

    Cost for 4 children = 4 × c = 4c

  3. 3

    Total cost = 2a + 4c

Recap

  • An expression is a collection of terms with no equals sign.
  • A variable is a letter representing an unknown number.
  • In algebra, we usually omit the multiplication sign (e.g., 4n instead of 4 × n).
  • Pay close attention to phrases like 'more than' (add) and 'less than' (subtract).
  • The number in front of a variable is called the coefficient.

Quick check

  1. Write an expression for '10 less than a number y'.1 mark
  2. What is the total cost of 'x' apples at 30p each and 'y' bananas at 25p each?1 mark

2. Plugging in the Numbers: Substitution

Substitution is the process of replacing variables in an expression or formula with their given numerical values. Once you've replaced the letters with numbers, you simply calculate the result. It is crucial to follow the correct order of operations (BIDMAS/BODMAS) to get the right answer.

BIDMAS: Brackets, Indices, Division, Multiplication, Addition, Subtraction

Key term

Substitution: The process of replacing variables in an expression or formula with their specific numerical values to evaluate it.

Common pitfall

Forgetting the order of operations. For example, in 3 + 2x where x=4, calculating 3+2 first to get 5×4=20, instead of the correct 3 + (2×4) = 3+8=11.

Fun fact

Substitution is used constantly in computer programming and spreadsheet software. When you type '=A1+B1' in a spreadsheet, the software substitutes the cell references with their values to calculate the result.

Worked example 12 marks

Evaluate 5x + 2y when x = 4 and y = 7.

  1. 1

    Write down the expression: 5x + 2y

  2. 2

    Substitute the values: x = 4 and y = 7.

  3. 3

    Calculation: 5(4) + 2(7)

  4. 4

    Follow BIDMAS (multiplication first): 20 + 14

  5. 5

    Final Answer: 34

Worked example 22 marks

The formula for the perimeter of a rectangle is P = 2(l + w). Find the perimeter if the length l is 8 cm and the width w is 5 cm.

  1. 1

    Write down the formula: P = 2(l + w)

  2. 2

    Substitute l = 8 and w = 5: P = 2(8 + 5)

  3. 3

    Follow BIDMAS (brackets first): P = 2(13)

  4. 4

    Calculate the final answer: P = 26

  5. 5

    The perimeter is 26 cm.

Recap

  • Substitution means replacing letters with their given number values.
  • Always use the order of operations (BIDMAS/BODMAS).
  • When substituting a negative number, use brackets to avoid errors, e.g., 3x with x=-2 becomes 3(-2).
  • A formula is a special type of equation that shows the relationship between different variables.

Quick check

  1. If a = 6 and b = 3, what is the value of 4a - 5b?2 marks
  2. Using the formula v = u + at, find v when u = 10, a = 3 and t = 5.2 marks

3. Tidying Up Algebra: Collecting Like Terms

Simplifying an expression means making it shorter and easier to work with. One way to do this is by 'collecting like terms'. Like terms are terms that have the exact same variable part (e.g., 3x and 5x are like terms, but 3x and 5y are not). You can think of it like collecting apples and bananas; you can count how many apples you have in total, and how many bananas, but you can't add them together to get 'apple-bananas'. To collect like terms, you add or subtract their coefficients (the numbers in front).

ax + bx = (a+b)x

Key term

Like Terms: Terms whose variables (and their powers) are exactly the same, such as 2x and 7x, or 3a²b and -5a²b.

Examiner insight

Examiners reward clear working where students group or highlight like terms before simplifying, as it demonstrates a systematic approach and reduces the chance of sign errors.

Common pitfall

Incorrectly handling signs. When simplifying 8x - 2y - 10x, students often forget the minus sign belongs to the 10x, leading to an incorrect x-term. The correct calculation is 8x - 10x = -2x.

Worked example 12 marks

Simplify the expression 7a + 4b - 2a + 5b.

  1. 1

    Identify the 'a' terms: +7a and -2a.

  2. 2

    Identify the 'b' terms: +4b and +5b.

  3. 3

    Group the like terms together: (7a - 2a) + (4b + 5b).

  4. 4

    Combine the 'a' terms: 7a - 2a = 5a.

  5. 5

    Combine the 'b' terms: 4b + 5b = 9b.

  6. 6

    Write the simplified expression: 5a + 9b.

Worked example 23 marks

Simplify 3xy + 8x - xy + 2x - 9.

  1. 1

    Identify the 'xy' terms: +3xy and -xy. Remember that -xy is -1xy.

  2. 2

    Identify the 'x' terms: +8x and +2x.

  3. 3

    Identify the number terms (constants): -9.

  4. 4

    Group them: (3xy - xy) + (8x + 2x) - 9.

  5. 5

    Combine them: 2xy + 10x - 9.

Recap

  • Like terms have the identical variable part.
  • To simplify, you combine the coefficients of like terms.
  • Always include the sign (+ or -) that is in front of a term when you group it.
  • Terms with different variables (e.g., x and y) cannot be combined.
  • Terms with different powers (e.g., x and x²) cannot be combined.

Quick check

  1. Simplify the expression 12p + 5q - 4p - 8q.2 marks

4. Multiplying Out: Expanding Single Brackets

Expanding brackets means to remove them by multiplying everything inside the bracket by the term directly outside it. This process is also known as the distributive law. A helpful way to remember this is the 'claw' method: draw arcs from the term outside to each term inside to make sure you don't miss any.

a(b + c) = ab + ac

a(b - c) = ab - ac

Key term

Expand: To remove brackets in an algebraic expression by multiplying each term inside the bracket by the term outside it.

Common pitfall

Only multiplying the first term inside the bracket. For 3(x + 5), a common error is to write 3x + 5 instead of the correct 3x + 15.

Fun fact

The distributive law, which is what we use to expand brackets, is a fundamental rule of numbers you use without thinking. To calculate 7 × 13 in your head, you might do (7 × 10) + (7 × 3) = 70 + 21 = 91. That's just expanding 7(10+3)!

Worked example 11 mark

Expand the expression 5(y + 3).

  1. 1

    The term outside the bracket is 5. The terms inside are y and +3.

  2. 2

    Multiply the outside term by the first term inside: 5 × y = 5y.

  3. 3

    Multiply the outside term by the second term inside: 5 × 3 = 15.

  4. 4

    Combine the results: 5y + 15.

Worked example 23 marks

Expand and simplify 4(2x - 3) - 2(x + 5).

  1. 1

    Expand the first bracket: 4 × 2x = 8x and 4 × (-3) = -12. This gives 8x - 12.

  2. 2

    Expand the second bracket, being careful with the sign: -2 × x = -2x and -2 × 5 = -10. This gives -2x - 10.

  3. 3

    Write out the full expanded expression: 8x - 12 - 2x - 10.

  4. 4

    Now collect the like terms: (8x - 2x) + (-12 - 10).

  5. 5

    Simplify to get the final answer: 6x - 22.

Recap

  • To expand a bracket, multiply the term on the outside by every term on the inside.
  • Be very careful with negative signs.
  • A negative number multiplied by a positive number gives a negative result.
  • A negative number multiplied by a negative number gives a positive result.
  • Sometimes you need to collect like terms after expanding to fully simplify the expression.

Quick check

  1. Expand the expression 6(m - 4).1 mark
  2. Expand -3(2a - b).2 marks

5. Finding the Unknown: Solving Simple Equations

An equation is a statement that two expressions are equal, shown by an equals sign (=). Solving an equation means finding the value of the unknown variable that makes the statement true. The key principle is to keep the equation 'balanced'. Whatever you do to one side of the equals sign, you must do the exact same thing to the other side. We use 'inverse operations' (the opposite operations) to isolate the variable on its own.

Inverse of Add (+) is Subtract (-)

Inverse of Subtract (-) is Add (+)

Inverse of Multiply (×) is Divide (÷)

Inverse of Divide (÷) is Multiply (×)

Key term

Equation: A mathematical statement that asserts the equality of two expressions, indicated by an equals sign (=).

Examiner insight

Examiners award marks for correctly setting up the equation from a word problem, as well as for each correct step in the solving process. Showing clear, logical working is essential to gain full marks.

Common pitfall

Performing operations in the wrong order. For a two-step equation like 2x + 5 = 11, you must subtract the 5 before dividing by 2. Trying to divide first is a common error.

Worked example 12 marks

Solve the equation 3y - 5 = 16.

  1. 1

    Write the equation: 3y - 5 = 16.

  2. 2

    Our goal is to get 'y' by itself. First, we undo the '- 5'.

  3. 3

    Add 5 to both sides: 3y - 5 + 5 = 16 + 5.

  4. 4

    This simplifies to: 3y = 21.

  5. 5

    Now, we undo the 'multiply by 3'.

  6. 6

    Divide both sides by 3: 3y / 3 = 21 / 3.

  7. 7

    This gives the final answer: y = 7.

Worked example 23 marks

I think of a number, multiply it by 4 and then add 3. The result is 23. Form an equation and solve it to find the number.

  1. 1

    Let the unknown number be 'n'.

  2. 2

    Translate the words into an equation: 'multiply by 4' is 4n, 'add 3' is + 3, 'result is 23' is = 23.

  3. 3

    The equation is: 4n + 3 = 23.

  4. 4

    Subtract 3 from both sides: 4n = 20.

  5. 5

    Divide both sides by 4: n = 5.

  6. 6

    The number is 5.

Recap

  • An equation has an equals sign; an expression does not.
  • To solve an equation, you must keep it balanced.
  • Whatever operation you perform on one side, you must perform on the other.
  • Use inverse operations to isolate the variable.
  • Deal with any addition or subtraction first, then multiplication or division.

Quick check

  1. Solve the equation 5p = 45.1 mark
  2. Solve the equation x/4 + 2 = 5.2 marks

End-of-chapter exercise

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

  1. Write an expression for the perimeter of an equilateral triangle with side length 's' cm.2 marks
  2. If a = 5, b = -2 and c = 10, evaluate the expression a(b + c).2 marks
  3. Simplify the expression 9p - 4q + 2r - 3p + 7q - r.3 marks
  4. Expand and simplify 6(y + 2) + 3(y - 4).3 marks
  5. Solve the equation 5x - 8 = 22.2 marks
  6. A plumber charges a £40 call-out fee plus £25 per hour. Write a formula for the total cost, C, for a job that takes h hours. Then, use the formula to find the cost of a 3-hour job.3 marks
  7. The perimeter of the rectangle shown is 46 cm. The sides are (2x + 3) cm and (x - 1) cm. Form an equation and solve it to find the value of x.4 marks
  8. Simplify fully: 3x²y - 7xy² + 5x²y + 4xy².3 marks
  9. Amir is x years old. His sister is 5 years older than him. The sum of their ages is 31. Write an equation in terms of x and solve it to find Amir's age.3 marks
  10. Expand and simplify -4a(2b - 3) - 5a.3 marks

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