Introduction to algebra

1. Overview

Algebra is the foundation of much of IGCSE mathematics. It uses letters (variables) to represent numbers, allowing us to express general mathematical relationships and solve problems. This topic covers the basics of algebraic expressions, including how to write them, simplify them, and find their numerical value by substituting numbers for variables. A solid understanding of these concepts is crucial for success in more advanced topics like solving equations, working with formulas, and graphing functions.

Key Definitions

• Variable: A letter (like $x, y,$ or $n$) used to represent a value that can change or is unknown.
• Expression: A mathematical phrase containing numbers, variables, and operators (e.g., $3x + 5$). It does not have an equals sign.
• Equation: A mathematical statement showing that two expressions are equal (e.g., $2x - 1 = 9$).
• Formula: A special type of equation that shows the relationship between different quantities (e.g., $A = lw$).
• Term: A single part of an expression, separated by $+$ or $-$ signs (e.g., in $4x - 7$, $4x$ and $7$ are terms).
• Coefficient: The number multiplying a variable (e.g., in $5x^2$, the coefficient is 5).

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Core Content

Letters as Generalised Numbers

In algebra, we use letters to represent numbers to create general rules.

• Addition: $a + a + a = 3a$ (Numerical example: $5 + 5 + 5 = 3 \times 5 = 15$)
• Multiplication: $a \times b$ is written as $ab$. We do not use the '$\times$' symbol as it can be confused with the letter $x$.
• Division: $a \div b$ is written as a fraction $\frac{a}{b}$.
• Powers: $x \times x$ is written as $x^2$.

Substitution

Substitution is the process of replacing a letter with a specific number to calculate the value of an expression.

Step-by-Step Method:

1. Replace every instance of the letter with the given number using parentheses (brackets), especially for negative numbers.
2. Follow the order of operations (BIDMAS: Brackets, Indices, Division/Multiplication, Addition/Subtraction).
3. Calculate the final result.

Worked example 1 — Simple Substitution

Question: Evaluate the expression $7x + 2$ when $x = 3$.

• Step 1 (Substitute): $7(3) + 2$
  • Reason: Replace $x$ with $3$ using parentheses.
• Step 2 (Multiply): $21 + 2$
  • Reason: Perform the multiplication.
• Step 3 (Add): $23$
  • Reason: Perform the addition.

Final Answer: $\boxed{23}$

Worked example 2 — Fraction Substitution

Question: Find the value of $\frac{5 + y}{y - 2}$ when $y = 8$.

• Step 1 (Substitute): $\frac{5 + (8)}{(8) - 2}$
  • Reason: Replace $y$ with $8$ using parentheses.
• Step 2 (Simplify numerator): $\frac{13}{(8) - 2}$
  • Reason: Perform the addition in the numerator.
• Step 3 (Simplify denominator): $\frac{13}{6}$
  • Reason: Perform the subtraction in the denominator.

Final Answer: $\boxed{\frac{13}{6}}$

Worked example 3 — Negative Substitution

Question: Evaluate $x^2 - 4x + 1$ when $x = -1$.

• Step 1 (Substitute): $(-1)^2 - 4(-1) + 1$
  • Reason: Replace $x$ with $-1$ using parentheses.
• Step 2 (Indices): $1 - 4(-1) + 1$
  • Reason: Evaluate $(-1)^2 = (-1) \times (-1) = 1$.
• Step 3 (Multiply): $1 + 4 + 1$
  • Reason: Evaluate $-4 \times (-1) = 4$.
• Step 4 (Add): $6$
  • Reason: Perform the addition.

Final Answer: $\boxed{6}$

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Extended Content (Extended Curriculum Only)

The Extended curriculum requires handling more complex substitution involving powers, roots, multiple variables, and more complex expressions. A strong understanding of the order of operations (BIDMAS/PEMDAS) is essential. You must also be comfortable working with negative numbers and fractions.

Worked example 4 — Powers and Negatives

Question: Find the value of $z = 2x^3 - x^2 + 4$ when $x = -3$.

• Step 1 (Substitute with brackets): $2(-3)^3 - (-3)^2 + 4$
  • Reason: Replace $x$ with $-3$ using parentheses.
• Step 2 (Indices first): $2(-27) - (9) + 4$
  • Reason: Evaluate $(-3)^3 = (-3) \times (-3) \times (-3) = -27$ and $(-3)^2 = (-3) \times (-3) = 9$.
• Step 3 (Multiply): $-54 - 9 + 4$
  • Reason: Perform the multiplication.
• Step 4 (Subtract/Add): $-59$
  • Reason: Perform the subtraction and addition.

Final Answer: $\boxed{-59}$

Worked example 5 — Square Roots and Fractions

Question: Find the value of $w = \sqrt{\frac{4ab}{c}}$ when $a = 5, b = 2, c = 4$.

• Step 1 (Substitute): $\sqrt{\frac{4(5)(2)}{4}}$
  • Reason: Replace $a$, $b$, and $c$ with their given values using parentheses.
• Step 2 (Multiply in numerator): $\sqrt{\frac{40}{4}}$
  • Reason: Evaluate $4 \times 5 \times 2 = 40$.
• Step 3 (Divide): $\sqrt{10}$
  • Reason: Evaluate $\frac{40}{4} = 10$.

Final Answer: $\boxed{\sqrt{10}}$

Worked example 6 — Multiple Variables and Grouping

Question: Evaluate $p = \frac{2(m+n)^2}{3k}$ when $m = 1, n = 4, k = 5$.

• Step 1 (Substitute): $p = \frac{2(1+4)^2}{3(5)}$
  • Reason: Replace $m$, $n$, and $k$ with their given values using parentheses.
• Step 2 (Simplify inside parentheses): $p = \frac{2(5)^2}{3(5)}$
  • Reason: Evaluate $1+4 = 5$.
• Step 3 (Evaluate exponent): $p = \frac{2(25)}{3(5)}$
  • Reason: Evaluate $5^2 = 25$.
• Step 4 (Multiply): $p = \frac{50}{15}$
  • Reason: Evaluate $2 \times 25 = 50$ and $3 \times 5 = 15$.
• Step 5 (Simplify fraction): $p = \frac{10}{3}$
  • Reason: Divide numerator and denominator by 5.

Final Answer: $\boxed{\frac{10}{3}}$

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Key Equations

While there are no fixed "Introduction to Algebra" formulas on the formula sheet, you must be able to use the following structures:

Concept	Structure	Notes
Expression	$ax + b$	$a$ and $b$ are constants; $x$ is the variable.
Power Expression	$ax^n$	Apply the power $n$ to $x$ before multiplying by $a$.
Fractional Form	$\frac{x+a}{b}$	The line acts as a bracket; solve the top before dividing.
General Formula	Subject = Expression	e.g., $A = lw$ (Area = length × width)

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Common Mistakes to Avoid

• ❌ Wrong (Incorrect Order of Operations): Calculating $5 + 2x$ as $7x$ when $x = 4$.
  • ✓ Right: Remember to multiply before adding: $5 + 2(4) = 5 + 8 = 13$.
• ❌ Wrong (Forgetting Negative Sign): Evaluating $-x^2$ as a positive number when $x$ is negative.
  • ✓ Right: Use parentheses to ensure the negative sign is included in the square: If $x = -2$, then $-x^2 = -(-2)^2 = -(4) = -4$.
• ❌ Wrong (Substituting Without Parentheses): Writing $3x + 1$ as $3-2+1$ when $x = -2$.
  • ✓ Right: Always use parentheses when substituting: $3(-2) + 1 = -6 + 1 = -5$.
• ❌ Wrong (Incorrect Fraction Evaluation): Calculating $\frac{a}{b+c}$ as $\frac{a}{b} + c$.
  • ✓ Right: The fraction bar acts as a grouping symbol. You must evaluate the entire denominator before dividing: $\frac{6}{2+1} = \frac{6}{3} = 2$.
• ❌ Wrong (Confusing Operations): Misinterpreting $4x$ as $4 + x$ instead of $4 \times x$.
  • ✓ Right: Remember that a number directly next to a variable implies multiplication.

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Exam Tips

• Command Words: "Evaluate" or "Find the value of" both mean you should substitute the numbers and give a final numerical answer.
• Show Your Substitution: Even if your final calculation is wrong, you often get a method mark for showing the numbers correctly placed into the expression.
• Parentheses are Mandatory: When substituting into a calculator, use the $(\ )$ buttons for every value you substitute. This prevents errors with negative signs and the order of operations.
• Units: If the substitution involves a real-world formula (e.g., $Area = lw$), ensure your final answer includes the correct units (e.g., $cm^2$).
• Non-Calculator Papers: If this appears on a non-calculator paper, the numbers will usually be simple integers or common fractions. Double-check your basic multiplication tables!
• Double-Check: After substituting, take a moment to visually inspect your expression to ensure you've replaced each variable correctly and haven't missed any signs.