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Recap A — Basic Concepts of Algebra

Substitution, the laws of indices and solving equations — explained simply with 15 worked examples.

The Big Idea

In algebra, a letter stands for a number . 3 x means 3 \times x, x y means x \times y, and x^{2} means x \times x . This topic has three skills: 1 Substitution — put numbers in place of the letters and work out the answer. 2 Laws of indices — simplify powers, such as x^{5} \div x^{2} . 3 Solving equations — find the value of x that makes an equation true.

1. Substitution

Example 1 — a simple expression

Question: Find 4 x + 7 when x = 3 . Answer: 4 (3) + 7 = 12 + 7 = 19

Example 2 — collect like terms first

Question: Find 6 x + 12 x - 5 when x = 2 . Answer: Like terms have the same letter: 6 x + 12 x = 18 x . 18 (2) - 5 = 36 - 5 = 31

Example 3 — with a power

Question: Find 2 x^{2} + 3 x when x = 5 . Answer: 1 Power first: 5^{2} = 25 2 2 (25) + 3 (5) = 50 + 15 = 65

Example 4 — a negative number

Question: Find 3 a - 2 b when a = 4 and b = - 1 . Answer: 3 (4) - 2 (- 1) = 12 + 2 = 14 Two minus signs together make a plus: - 2 \times - 1 = + 2 .

Example 5 — a fraction (like the paper)

Question: Find \frac{( 3 x ^{2} \cdot 2 y ) + 6}{x + y} when x = 1, y = 2 . Answer: Top: 3 (1)^{2} \times 2 (2) + 6 = 3 \times 4 + 6 = 12 + 6 = 18 Bottom: 1 + 2 = 3 Answer: 18 \div 3 = 6

Example 6 — another fraction

Question: Find \frac{x ^{2} + y ^{2}}{x - y} when x = 4, y = 2 . Answer: Top: 16 + 4 = 20 Bottom: 4 - 2 = 2 Answer: 20 \div 2 = 10

2. The Laws of Indices

An index (power) tells you how many times to multiply: x^{3} = x \times x \times x . Rule In words Example x^{a} \times x^{b} = x^{a + b} multiply \to add the indices x^{4} \times x^{3} = x^{7} x^{a} \div x^{b} = x^{a - b} divide \to subtract the indices x^{7} \div x^{3} = x^{4} (x^{a})^{b} = x^{ab} power of a power \to multiply the indices (x^{2})^{4} = x^{8} x^{0} = 1 anything (except 0) to the power 0 is 1 9^{0} = 1

Rule	In words	Example
$x^{a} \times x^{b} = x^{a + b}$	multiply → add the indices	$x^{4} \times x^{3} = x^{7}$
$x^{a} \div x^{b} = x^{a - b}$	divide → subtract the indices	$x^{7} \div x^{3} = x^{4}$
$(x^{a})^{b} = x^{ab}$	power of a power → multiply the indices	$(x^{2})^{4} = x^{8}$
$x^{0} = 1$	anything (except $0$ ) to the power $0$ is $1$	$9^{0} = 1$

Example 1 — divide

Question: Simplify \frac{x ^{7}}{x ^{3}} . Answer: Divide \to subtract: x^{7 - 3} = x^{4}

Example 2 — numbers and letters together

Question: Simplify 3 x (2 y) (4 x y) . Answer: 1 Multiply the numbers : 3 \times 2 \times 4 = 24 2 Count each letter : x \times x = x^{2} and y \times y = y^{2} Answer: 24 x^{2} y^{2}

Example 3 — powers of the same letter

Question: Simplify 5 a^{2} (2 a^{3}) . Answer: Numbers: 5 \times 2 = 10 . Letters: a^{2 + 3} = a^{5} . Answer: 10 a^{5}

Example 4 — divide numbers and letters

Question: Simplify \frac{12 x ^{6} y ^{4}}{4 x ^{2} y} . Answer: Numbers: 12 \div 4 = 3 x : x^{6 - 2} = x^{4} y : y^{4 - 1} = y^{3} (on its own, y means y^{1}) Answer: 3 x^{4} y^{3}

Example 5 — power of a power

Question: Simplify (x^{3})^{2} \times x . Answer: (x^{3})^{2} = x^{3 \times 2} = x^{6} . Then x^{6} \times x^{1} = x^{7} .

3. Solving Equations

Example 1 — given y

Question: Solve 5 x - 3 y = 11, given y = 3 . Answer: 1 Put in y = 3 : 5 x - 9 = 11 2 Add 9 to both sides: 5 x = 20 3 Divide both sides by 5 : x = 4 Check: 5 (4) - 3 (3) = 20 - 9 = 11 ✓

Example 2 — a fraction (like the paper)

Question: Solve , given .Answer:1Right side: 2Bottom of the fraction: , so 3Multiply both sides by : 4Subtract : , so Check: ✓

Example 3 — brackets on one side

Question: Solve 2 (x + 3) = 4 x - 8 . Answer: 1 Expand: 2 x + 6 = 4 x - 8 2 Subtract 2 x from both sides: 6 = 2 x - 8 3 Add 8 : 14 = 2 x, so x = 7 Check: 2 (10) = 20 and 4 (7) - 8 = 20 ✓

Your turn — try these on paper, then check the answers below. 1 . Find 3 x + 9 x - 4 when x = 2 . 2 . Simplify a^{8} \div a^{5} . 3 . Simplify 2 x^{2} (3 y) (x y) . 4 . Solve 4 x + 2 y = 30, given y = 5 .

Example 1 — Answers to Your Turn

1 . 12 x - 4 = 24 - 4 = 20 2 . a^{8 - 5} = a^{3} 3 . 2 \times 3 = 6; x^{2 + 1} = x^{3}; y \times y = y^{2} \to 6 x^{3} y^{2} 4 . 4 x + 10 = 30 \Rightarrow 4 x = 20 \Rightarrow x = 5

Next: Recap B — Set Operations

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