10.4 Algebraic Expressions, Linear Equations & Word Problems

Algebraic Expressions, Linear Equations & Word Problems

Core Concept: In the Edutest Mathematics subtest, algebra is not an isolated academic topic—it is the universal language and computational engine that powers numerical reasoning, geometric deduction, and complex word problems. Strong results require simplifying multi-variable expressions, manipulating index laws, clearing fractional equations in one step, and translating multi-sentence word problems into solvable linear models quickly.

While some students attempt to solve scholarship word problems by testing the multiple-choice options (back-solving) or through numerical trial-and-error, this approach collapses when questions feature large numbers, fractions, or negative solutions. Setting up crisp, single-variable linear equations guarantees a direct, unambiguous path to the correct answer while banking essential minutes on the exam clock.

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Algebraic Vocabulary & Structural Building Blocks

Precise understanding of algebraic anatomy prevents fundamental operational errors:

                         Anatomy of an Algebraic Expression
                                  3x² - 5x + 7
                                  ││    │    │
     Coefficient of x² ───────────┘│    │    └── Constant Term (no variable)
     Variable Base with Index ─────┘    └── Term with Coefficient of -5

• Variable: A letter or symbol representing an unknown numerical value (e.g., $x, y, n$).
• Term: A single number, variable, or product of numbers and variables separated by $+$ or $-$ signs (e.g., in $4x^2 - 7x + 9$, the terms are $4x^2$, $-7x$, and $+9$).
• Coefficient: The numerical multiplier positioned immediately before a variable (e.g., in $-8x$, the coefficient is $-8$; in $x$, the implicit coefficient is $+1$).
• Constant: A standalone number with no variable attached (e.g., $+7$).
• Expression vs Equation: An expression is a mathematical phrase without an equals sign ($4x - 9$); it can only be simplified or evaluated. An equation asserts equality between two expressions ($4x - 9 = 15$); it can be solved for the unknown.
• Like Terms: Terms that have the exact same variable base raised to the exact same exponents, regardless of their numerical coefficients. Only like terms can be combined additively:
  • $5x^2y$ and $-2x^2y$ are like terms (sum: $3x^2y$).
  • $4xy^2$ and $4x^2y$ are not like terms (different variable exponents).

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Index Laws (The Laws of Exponents)

In expressions with powers, indices follow strict structural rules. Always distinguish between operations performed on coefficients (which follow ordinary arithmetic) and operations performed on indices (which follow index laws).

                              The Four Essential Index Laws
  1. Multiplication Law:    xᵃ · xᵇ = xᵃ⁺ᵇ          Keep base, ADD indices
  2. Division Law:          xᵃ ÷ xᵇ = xᵃ⁻ᵇ          Keep base, SUBTRACT indices
  3. Power of a Power Law:  (xᵃ)ᵇ = xᵃᵇ             Keep base, MULTIPLY indices
  4. Zero Power Law:        x⁰ = 1  (x ≠ 0)         Any non-zero base to power 0 equals 1

Extended Index Rules

• Power of a Product: $(ab)^n = a^n b^n$. When a coefficient is enclosed inside brackets, it must also be raised to the power: $(3x^4)^3 = 3^3 \times (x^4)^3 = 27x^{12} \quad (\text{NOT } 9x^{12} \text{ or } 3x^{12})$
• Negative Index Law: A negative exponent denotes the reciprocal of the base raised to the positive power: $x^{-n} = \frac{1}{x^n}, \quad \left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ $4^{-2} = \frac{1}{4^2} = \frac{1}{16}, \quad \left(\frac{2}{3}\right)^{-3} = \left(\frac{3}{2}\right)^3 = \frac{27}{8}$

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Expansion & Factorisation

Expansion (removing brackets) and factorisation (inserting brackets) are inverse algebraic processes.

1. The Distributive Law (Single Brackets)

Multiply the term outside the brackets by every individual term inside:

$a(b + c) = ab + ac, \quad a(b - c) = ab - ac$

The Negative Sign Checkpoint: When multiplying by a negative term outside the brackets, every sign inside flips: $-4(3x - 5) = -12x + 20 \quad (\text{NOT } -12x - 20)$

2. Binomial Expansion (Double Brackets - FOIL)

To expand $(a + b)(c + d)$, apply FOIL (First, Outer, Inner, Last):

$(a + b)(c + d) = \underbrace{ac}_{\text{First}} + \underbrace{ad}_{\text{Outer}} + \underbrace{bc}_{\text{Inner}} + \underbrace{bd}_{\text{Last}}$

Worked Example: Expand and simplify $(2x - 3)(4x + 5)$:

• First: $2x \times 4x = 8x^2$
• Outer: $2x \times 5 = +10x$
• Inner: $-3 \times 4x = -12x$
• Last: $-3 \times 5 = -15$
• Combine like terms: $8x^2 + 10x - 12x - 15 = 8x^2 - 2x - 15$

3. Special Algebraic Products

• Perfect Squares: $(a + b)^2 = a^2 + 2ab + b^2$ $(a - b)^2 = a^2 - 2ab + b^2$
• Difference of Two Squares (DOTS): $(a - b)(a + b) = a^2 - b^2$

4. Factorisation: Highest Common Factor (HCF)

To factorise an expression, identify the largest numerical factor and the highest common power of each variable present in every term:

$18x^3y^2 - 24x^2y^3 = 6x^2y^2(3x - 4y)$

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Solving Linear Equations

Solving a linear equation means isolating the unknown variable on one side of the equals sign using inverse operations applied in reverse order of operations (SADMEP: Subtraction/Addition first, then Division/Multiplication).

1. Variables on Both Sides

Collect all variable terms on the side with the larger positive coefficient to keep the variable term positive, and collect all constant numbers on the opposite side:

$9x - 14 = 4x + 21 \implies 9x - 4x = 21 + 14 \implies 5x = 35 \implies x = 7$

2. Equations with Grouping Brackets

Expand all brackets, collect like terms on each side independently, and then isolate the variable:

$3(4x - 2) - 2(x + 5) = 24 \implies 12x - 6 - 2x - 10 = 24 \implies 10x - 16 = 24 \implies 10x = 40 \implies x = 4$

3. Fractional Linear Equations: The LCD Clearing Method

Never work with cumbersome fraction additions across multiple steps. Find the Lowest Common Denominator (LCD) of all fractions in the equation and multiply every single term on both sides by that LCD to eliminate all denominators in one stroke.

Worked Example: Multi-Term Fractional Equation

$\text{Solve: } \frac{3x + 1}{2} - \frac{2x - 1}{3} = 5$

• Step 1 (Identify LCD): Denominators are 2 and 3 $\implies \text{LCD} = 6$.
• Step 2 (Multiply EVERY term by 6): $6 \times \left(\frac{3x + 1}{2}\right) - 6 \times \left(\frac{2x - 1}{3}\right) = 6 \times 5$
• Step 3 (Simplify each term): $3(3x + 1) - 2(2x - 1) = 30$
• Step 4 (Expand brackets carefully): $9x + 3 - 4x + 2 = 30$
• Step 5 (Combine like terms and solve): $5x + 5 = 30 \implies 5x = 25 \implies x = 5$

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Algebraic Substitution Protocols

Evaluating expressions by replacing variables with given numerical values is heavily tested on scholarship papers. The primary source of error is squaring negative numbers.

The Negative Squaring Rule:

• $(-x)^2 = (-x) \times (-x) = +x^2$ (The bracket encloses the negative sign $\implies$ Result is positive).
• $-x^2 = -(x^2) = -(x \times x) = -x^2$ (No bracket $\implies$ Square first, then negate).

Worked Example: Evaluate $E = 4a^2 - 3ab + b^2$ when $a = -3$ and $b = 4$:

• $4a^2 = 4(-3)^2 = 4(9) = 36$
• $-3ab = -3(-3)(4) = -3(-12) = +36$
• $b^2 = (4)^2 = 16$
• Total: $E = 36 + 36 + 16 = 88$

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Translating Word Problems into Linear Equations

The Verbal-to-Algebraic Translation Guide

Classic Word Problem Archetypes

Archetype 1: Consecutive Integers

"The sum of three consecutive integers is 72. What is the largest integer?"

• Let the integers be $n, n+1, n+2$.
• Setup: $n + (n+1) + (n+2) = 72 \implies 3n + 3 = 72 \implies 3n = 69 \implies n = 23$.
• Largest integer: $n + 2 = 23 + 2 = 25$.

Archetype 2: Age Problems (The Dual-Timeline Table)

Age problems involve establishing relationships between two people across different points in time (now vs past or future).

"Marcus is currently 3 times as old as his younger sister Chloe. In 8 years, Marcus will be twice as old as Chloe will be then. How old is Marcus right now?"

  Timeline Table:           Current Age        Age in 8 Years (+8)
  ─────────────────────────────────────────────────────────────────
  Chloe                     c                  c + 8
  Marcus                    3c                 3c + 8

• Setup equation using the future relationship: $\text{Marcus in 8 yrs} = 2 \times (\text{Chloe in 8 yrs})$ $3c + 8 = 2(c + 8)$
• Solve for $c$: $3c + 8 = 2c + 16 \implies 3c - 2c = 16 - 8 \implies c = 8$
• Check the Question Prompt: Chloe is 8 years old right now. The question asks for Marcus's age: $\text{Marcus} = 3c = 3(8) = 24\text{ years old}$.

Archetype 3: Geometric Dimension Relationships

"The length of a rectangular sports ground is $6\text{ metres}$ longer than three times its width. If the total perimeter is $140\text{ metres}$, find the length."

• Let width $= w$. Length $l = 3w + 6$.
• Perimeter formula: $P = 2(l + w) = 140 \implies l + w = 70$.
• Substitute: $(3w + 6) + w = 70 \implies 4w + 6 = 70 \implies 4w = 64 \implies w = 16\text{ m}$.
• Length $l = 3(16) + 6 = 48 + 6 = 54\text{ metres}$.

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High-Yield Traps in Algebra & Word Problems

1. Answering the Variable Instead of the Prompt

In word problems, candidates frequently solve for the unknown variable (e.g., $w = 16$ or $c = 8$) and immediately select it from the options, forgetting that the question specifically requested the length ($54\text{ m}$) or the older person's age ($24$). Always re-read the final sentence of the question before bubbling your answer.

2. Forgetting to Multiply the Constant when Clearing Fractions

When multiplying an equation by the LCD, students often multiply the fractional terms but forget to multiply a standalone constant on the right-hand side: $\frac{x}{2} + \frac{x}{3} = 5 \implies 3x + 2x = 5 \quad (\text{FATAL ERROR!})$ The right-hand side must also be multiplied by 6: $3x + 2x = 30 \implies 5x = 30 \implies x = 6$.

3. One-Sided Age Shifts

When writing equations for age problems set 5 years in the future, students often add 5 years to the parent but forget to add 5 years to the child ($M + 5 = 2C$ instead of $M + 5 = 2(C + 5)$). Time advances equally for everyone!