Worked example 1 · Easy
Worked method
Write “seven times \(m\), plus four” as an expression.
Seven times \(m\) is \(7m\); add four: \(\boxed{7m+4}\).
Chapter 19
Introduction to algebra
Expressions, notation and accurate substitution. Original explanations and examination-focused practice based on the supplied textbook's topic sequence.
By the end of this chapter, you should be able to
- Explain the key ideas and use correct mathematical terminology in each textbook subtopic.
- Use the appropriate methods and show enough working for examination marks.
- Check intermediate arithmetic, signs, units and the final answer.
1
Subtopic 1
Letters for unknowns
Letters can represent numbers that are unknown or that change from situation to situation. A formula gives a rule that works for many numerical inputs, rather than only one example.
Write \(4\times a\) as \(4a\), \(b\times c\) as \(bc\), and \(k\div3\) as \(\frac{k}{3}\). By convention the numerical coefficient comes before the letter. State what your letters represent.
Be careful with notation: \(2x\) means two times \(x\), but \(x^2\) means \(x\times x\). Write brackets when multiplying a whole sum, for example \(3(a+b)\).
Rules and important ideas
- \(W=Nh\) means wage = hours × hourly rate.
- \(P=2l+2w\) gives the perimeter of a rectangle.
Worked example 2 · Medium
Worked method
A taxi charges £4 plus £2 for each mile. Write the fare \(F\) for \(d\) miles.
The distance charge is \(2d\). Add the fixed charge: \(\boxed{F=4+2d}\).
Worked example 3 · Exam-style
Worked method
A garden is \(3x\) metres long and \((x+2)\) metres wide. Write its perimeter without expanding the bracket.
\(P=2(3x)+2(x+2)=\boxed{6x+2(x+2)}\) metres. It can be simplified to \(8x+4\) m.
Exam technique
Identify what is given, write the correct rule, show the substitution or algebraic steps, and check the final result before moving on.
2
Subtopic 2
Substituting numbers into algebraic expressions
Substitute by replacing every instance of a letter with its given value. Work through brackets and powers before multiplying, dividing, adding or subtracting (BIDMAS).
Always surround substituted negative values with brackets. If \(x=-4\), then \(x^2=(-4)^2=16\), whereas \(-x^2=-(16)=-16\).
When an expression contains \(3x^2\), square \(x\) first and then multiply by 3; this is not \((3x)^2\).
Rules and important ideas
- \(x=-3:\quad 2x^2=2(-3)^2=18\)
- \(3(x+2)\) means multiply the entire bracket by 3.
Worked example 4 · Easy
Worked method
Calculate \(5a-3\) when \(a=4\).
\(5(4)-3=20-3=\boxed{17}\).
Worked example 5 · Medium
Worked method
Calculate \(2t^2-3t\) when \(t=-2\).
\(2(-2)^2-3(-2)=8+6=\boxed{14}\).
Worked example 6 · Exam-style
Worked method
A rectangle has area \(A=lw\). Find \(A\) when \(l=7.5\) cm and \(w=3.2\) cm.
\(A=7.5\times3.2=\boxed{24\text{ cm}^2}\).
Exam technique
Identify what is given, write the correct rule, show the substitution or algebraic steps, and check the final result before moving on.
3
Subtopic 3
Using harder numbers when substituting
Substitution also works with fractions, negative numbers and decimals. Write the substituted expression carefully, then evaluate without rounding too early.
For fractions, keep exact arithmetic when the question requests an exact answer: multiply numerator and denominator, use common denominators for addition, and simplify at the end.
Applied formulas often carry units. Check that inputs are in compatible units and that the final answer uses the correct one. Negative numbers in squared terms require brackets.
Rules and important ideas
- \(\left(\frac34\right)^2=\frac9{16}\)
- \(x=-0.2\Rightarrow x^2=0.04\).
Worked example 7 · Easy
Worked method
Find \(2x^2+3\) when \(x=0.5\).
\(2(0.5)^2+3=2(0.25)+3=\boxed{3.5}\).
Worked example 8 · Medium
Worked method
Find \(P=ab+2b^2\) when \(a=\frac12\), \(b=\frac34\).
\(P=\frac12\frac34+2\left(\frac34\right)^2=\frac38+\frac{18}{16}=\frac38+\frac98=\boxed{\frac32}\).
Worked example 9 · Exam-style
Worked method
The distance fallen is \(s=\frac12gt^2\). With \(g=9.8\) and \(t=2.5\), calculate \(s\).
\(s=\frac12(9.8)(2.5)^2=4.9(6.25)=\boxed{30.625\text{ m}}\).
Exam technique
Identify what is given, write the correct rule, show the substitution or algebraic steps, and check the final result before moving on.
Chapter 19 practice
These are newly written practice questions, not past-paper reproductions. Questions progress from confidence-building to exam-style. Answers are hidden until you open each question’s Show Answer section.
Letters for unknowns
1Confidence practice[2 marks]
Write \(9\times x\) using algebra notation.
Show Answer
Step-by-step solution: \(\boxed{9x}\).
2Confidence practice[2 marks]
Write “\(y\) divided by five” algebraically.
Show Answer
Step-by-step solution: \(\boxed{\frac y5}\).
3Developing practice[3 marks]
A concert ticket costs £12. Write a formula for the total cost \(C\) of \(n\) tickets.
Show Answer
Step-by-step solution: \(\boxed{C=12n}\).
4Developing practice[3 marks]
The sum of three consecutive whole numbers starts at \(n\). Write an expression for the sum.
Show Answer
Step-by-step solution: Numbers: \(n,n+1,n+2\). Sum \(=3n+3\).
5Exam-style practice[4 marks]
A rectangular field has length \(2x+3\) metres and width \(x\) metres. Write a formula for its perimeter.
Show Answer
Step-by-step solution: \(P=2(2x+3)+2x=\boxed{6x+6}\) metres.
6Exam-style practice[4 marks]
A cinema sells adult tickets for £a and child tickets for £c. Express the revenue from 8 adult and 11 child tickets.
Show Answer
Step-by-step solution: \(8\times a+11\times c=\boxed{8a+11c}\).
Substituting numbers into algebraic expressions
7Confidence practice[2 marks]
Find \(4x+7\) when \(x=3\).
Show Answer
Step-by-step solution: \(4(3)+7=\boxed{19}\).
8Confidence practice[2 marks]
Find \(a^2+b\) when \(a=5\) and \(b=3\).
Show Answer
Step-by-step solution: \(5^2+3=25+3=\boxed{28}\).
9Developing practice[3 marks]
Find \(3x^2+4\) when \(x=-3\).
Show Answer
Step-by-step solution: \(3(-3)^2+4=27+4=\boxed{31}\).
10Developing practice[3 marks]
Calculate \(P=2l+2w\) for \(l=4.5\), \(w=2.8\).
Show Answer
Step-by-step solution: \(P=2(4.5)+2(2.8)=9+5.6=\boxed{14.6}\).
11Exam-style practice[4 marks]
Calculate \(2a(3b-c)\) when \(a=4\), \(b=-2\), \(c=5\).
Show Answer
Step-by-step solution: \(3b-c=3(-2)-5=-11\). Then \(2(4)(-11)=\boxed{-88}\).
12Exam-style practice[4 marks]
Find the value of \(\dfrac{3a^2-b}{2}\) when \(a=-2\), \(b=-4\).
Show Answer
Step-by-step solution: \(\frac{3(-2)^2-(-4)}{2}=\frac{12+4}{2}=\boxed{8}\).
Using harder numbers when substituting
13Confidence practice[2 marks]
Calculate \(x^2+2x\) when \(x=0.5\).
Show Answer
Step-by-step solution: \((0.5)^2+2(0.5)=0.25+1=\boxed{1.25}\).
14Confidence practice[2 marks]
Calculate \(3a\) when \(a=\frac25\).
Show Answer
Step-by-step solution: \(3\times\frac25=\boxed{\frac65}\).
15Developing practice[3 marks]
Find \(2u^2-3v\) for \(u=-1.5\) and \(v=-2\).
Show Answer
Step-by-step solution: \(2(2.25)-3(-2)=4.5+6=\boxed{10.5}\).
16Developing practice[3 marks]
Find \(\dfrac{a+b}{c}\) for \(a=\frac12\), \(b=\frac34\), \(c=\frac58\).
Show Answer
Step-by-step solution: Numerator \(=\frac54\). Divide by \(\frac58\): \(\frac54\times\frac85=\boxed{2}\).
17Exam-style practice[4 marks]
A cuboid has surface area \(S=2lw+2lh+2wh\). Find \(S\) when \(l=4\), \(w=2.5\), \(h=3\), in cm.
Show Answer
Step-by-step solution: \(S=2(4)(2.5)+2(4)(3)+2(2.5)(3)=20+24+15=\boxed{59\text{ cm}^2}\).
18Exam-style practice[4 marks]
Calculate \(E=\dfrac{3x^2}{2a}\) for \(x=1.2\), \(a=0.8\).
Show Answer
Step-by-step solution: \(E=\frac{3(1.2)^2}{2(0.8)}=\frac{4.32}{1.6}=\boxed{2.7}\).
High-grade revision checklist
- I can explain each rule in the exact textbook topic order.
- I can select and apply the correct method, including signs, powers and brackets.
- I can show clear working in harder multi-step questions.
- I can check by reversing the operation, estimating, or substituting a value, as appropriate.