2.4 Solving Linear Equations, Expressions & Algebraic Word Problems

2.4 Solving Linear Equations, Expressions & Algebraic Word Problems

Algebraic reasoning is a critical skill for law enforcement officers in the Jamaica Constabulary Force (JCF). Whether determining the unit cost of tactical gear, calculating resource allocations across divisions, or solving inventory discrepancies in an evidence room, officers must translate real-world scenarios into mathematical expressions and solve linear equations accurately.

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1. Fundamentals of Algebraic Expressions & Linear Equations

An algebraic expression combines numbers, variables (such as $x$ or $y$), and operational symbols (addition, subtraction, multiplication, division). Unlike an equation, an expression does not contain an equals sign ($=$).

A linear equation is an algebraic statement declaring that two expressions are equal, where the variable is raised to the first power (e.g., $x^1$). The general form of a single-variable linear equation is:

$ax + b = c$

where $a$, $b$, and $c$ are known real numbers, and $x$ is the unknown variable to be solved.

Core Properties & Inverse Operations

To solve a linear equation, you must isolate the variable on one side of the equals sign by applying inverse operations (undoing operations in reverse order):

• Addition and subtraction are inverse operations.
• Multiplication and division are inverse operations.

Whatever operation is performed on one side of the equation must be performed on the other side to maintain balance.

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2. Simplifying Algebraic Expressions

Before solving complex linear equations, you must simplify expressions by combining like terms and applying the distributive property.

Combining Like Terms

Like terms are terms that contain the exact same variable raised to the exact same power. Only coefficients (the numbers in front of variables) are added or subtracted.

$\text{Example: } 7x + 4y - 3x + 9y + 5 = (7x - 3x) + (4y + 9y) + 5 = 4x + 13y + 5$

The Distributive Property

The distributive property allows you to multiply a single term across terms inside parentheses:

$a(b + c) = ab + ac$

$\text{Example: } 3(2x + 5) - 4(x - 2) = 6x + 15 - 4x + 8 = 2x + 23$

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3. Solving Multi-Step Linear Equations

Solving multi-step equations involves a systematic sequence of steps:

1. Clear parentheses using the distributive property.
2. Combine like terms on each side of the equation independently.
3. Move variable terms to one side by adding or subtracting variables.
4. Move constant terms to the opposite side using inverse addition or subtraction.
5. Isolate the variable by dividing or multiplying by its coefficient.
6. Verify your solution by substituting the result back into the original equation.

Worked Example 1: Solving $ax + b = cx + d$

Problem: Solve the equation $5x - 14 = 2x + 16$.

Step-by-Step Solution:

• Step 1: Subtract $2x$ from both sides to gather variables on the left side: $5x - 2x - 14 = 16 \implies 3x - 14 = 16$
• Step 2: Add $14$ to both sides to gather constants on the right side: $3x = 16 + 14 \implies 3x = 30$
• Step 3: Divide both sides by the coefficient $3$: $x = \frac{30}{3} \implies x = 10$
• Step 4: Check: $5(10) - 14 = 50 - 14 = 36$. On the right side: $2(10) + 16 = 20 + 16 = 36$. Both sides match ($36 = 36$).

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4. Translating Word Problems into Equations

JCF Entrance Examination math problems often present scenarios in word format. Translating written phrases into mathematical expressions is the key to solving them.

Common Translation Key

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5. Police Operational Math Scenarios

Scenario 1: Tactical Equipment Procurement

Problem: The St. Andrew South Police Division receives a total budget grant of $540,000 JMD to purchase tactical boots and high-visibility vests. Each pair of tactical boots costs $18,000 JMD, and each vest costs $6,000 JMD. The divisional logistics officer orders $3$ times as many vests as pairs of boots. How many pairs of boots and how many vests were purchased?

Step-by-Step Solution:

• Step 1: Define variables. Let $b$ equal the number of pairs of boots purchased.
• Step 2: Express other quantities. Since the officer buys $3$ times as many vests as boots, the number of vests is $3b$.
• Step 3: Set up the cost equation. $\text{Total Cost} = (\text{Cost per Boot} \times b) + (\text{Cost per Vest} \times 3b)$ $540,000 = 18,000b + 6,000(3b)$
• Step 4: Simplify the equation. $540,000 = 18,000b + 18,000b$ $540,000 = 36,000b$
• Step 5: Solve for $b$. $b = \frac{540,000}{36,000} = 15 \text{ pairs of boots}$
• Step 6: Calculate vests. $\text{Number of vests} = 3b = 3 \times 15 = 45 \text{ vests}$
• Step 7: Verification. Boots cost: $15 \times 18,000 = 270,000$. Vests cost: $45 \times 6,000 = 270,000$. Total cost: $270,000 + 270,000 = 540,000$ JMD.

Scenario 2: Station Inventory Audit

Problem: An inventory check at the Spanish Town Police Station evidence vault records a total of $140$ item containers comprising drug exhibit bags and property storage boxes. The number of drug exhibit bags is $20$ less than triple the number of property storage boxes. Find the number of drug exhibit bags.

Step-by-Step Solution:

• Step 1: Define variables. Let $p$ equal the number of property storage boxes.
• Step 2: Express drug exhibit bags in terms of $p$. $\text{Drug exhibit bags} = 3p - 20$
• Step 3: Set up total sum equation. $\text{Total items} = p + (3p - 20) = 140$
• Step 4: Solve for $p$. $4p - 20 = 140$ $4p = 160 \implies p = 40 \text{ property storage boxes}$
• Step 5: Calculate drug exhibit bags. $\text{Drug exhibit bags} = 3(40) - 20 = 120 - 20 = 100 \text{ bags}$
• Step 6: Verification. $40 + 100 = 140$ total items.