5.1 Linear Equations, Inequalities & Simultaneous Systems

5.1 Linear Equations, Inequalities & Simultaneous Systems

Operational command in the Ghana Armed Forces (GAF) demands rigorous quantitative competence. An officer cadet must calculate convoy transit times, allocate munitions under strict cargo payload constraints, and balance troop rations during field deployments. The foundation of all applied quantitative reasoning rests upon algebra: the language of symbolic relationships, constraints, and balance. In the GAF Officer Cadet Written Examination, algebra problems assess your speed, precision, and logical discipline. Mastery requires moving beyond rote memorization to understand the structural properties of linear equalities, simultaneous systems, and directional inequalities.

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Foundations of Algebraic Expressions & Simplification

An algebraic expression is a mathematical phrase combining numbers (constants), letters representing unknown quantities (variables), and arithmetic operations ($+, -, \times, \div$). Unlike an equation, an expression contains no equality sign ($=$) and cannot be "solved" for a specific numerical value; it can only be simplified, expanded, or evaluated for assigned values.

Anatomical Elements of an Expression

Consider the expression $7x^2 - 4xy + 9$:

• Terms: The distinct parts separated by addition or subtraction operators. Here, the terms are $7x^2$, $-4xy$, and $+9$.
• Variables: The alphabetic symbols representing unspecified quantities ($x$ and $y$).
• Coefficients: The numerical multipliers directly preceding the variables. The coefficient of $x^2$ is $7$; the coefficient of $xy$ is $-4$.
• Constants: Standalone numerical values that remain invariant ($+9$).
• Like Terms: Terms containing identical variable bases raised to identical exponents (e.g., $5x$ and $-12x$, or $3a^2b$ and $8a^2b$). Only like terms can be combined through addition or subtraction.

The Distributive Property and Nested Brackets

Simplifying complex expressions requires systematic removal of grouping symbols using the distributive law: $a(b + c) = ab + ac$. When expressions feature nested grouping symbols—parentheses $( )$, square brackets $[ ]$, and braces $\{ \}$—always expand from the innermost group outward.

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Solving Single-Variable Linear Equations

A linear equation in one variable can be expressed in the canonical form $ax + b = 0$, where $a, b \in \mathbb{R}$ and $a \neq 0$. Because the variable is raised exclusively to the first power ($x^1$), the equation yields exactly one unique real solution: $x = -\frac{b}{a}$.

Systematic 5-Step Solution Protocol

1. Clear Fractions and Decimals: Multiply every term on both sides of the equation by the Lowest Common Multiple (LCM) of all denominators.
2. Expand All Brackets: Apply the distributive property to eliminate all parentheses and brackets.
3. Combine Like Terms: Simplify expressions independently on the left-hand side (LHS) and right-hand side (RHS).
4. Isolate the Variable Term: Use addition or subtraction to collect all variable terms on one side and all constant terms on the opposite side.
5. Isolate the Variable: Multiply or divide by the variable's coefficient to obtain $x = k$, then substitute back into the original equation to verify.

Worked Example 1: Multi-Step Fractional Linear Equation

Solve the following linear equation for $x$:

$\frac{2x + 1}{5} - \frac{x - 1}{3} = \frac{x - 3}{4}$

Step 1: Determine the LCM of the denominators. The denominators are 5, 3, and 4. Since 5 and 3 are prime, $\text{LCM}(5, 3, 4) = 5 \times 3 \times 4 = 60$.

Step 2: Multiply every term on both sides by 60.

$60 \left( \frac{2x + 1}{5} \right) - 60 \left( \frac{x - 1}{3} \right) = 60 \left( \frac{x - 3}{4} \right)$

$12(2x + 1) - 20(x - 1) = 15(x - 3)$

Step 3: Expand the brackets carefully.

$24x + 12 - 20x + 20 = 15x - 45$

Notice that $-20 \times (-1) = +20$. Sign errors here invalidate all subsequent calculations.

Step 4: Combine like terms on each side.

$(24x - 20x) + (12 + 20) = 15x - 45$

$4x + 32 = 15x - 45$

Step 5: Transpose terms to isolate $x$.

$32 + 45 = 15x - 4x$

$77 = 11x$

$x = \frac{77}{11} = 7$

Verification Check: Substitute $x = 7$ into the original LHS: $\frac{2(7) + 1}{5} - \frac{7 - 1}{3} = \frac{15}{5} - \frac{6}{3} = 3 - 2 = 1$. RHS: $\frac{7 - 3}{4} = \frac{4}{4} = 1$. Since $\text{LHS} = \text{RHS} = 1$, the solution $x = 7$ is correct.

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Simultaneous Linear Systems with Two Variables

Military operations frequently present scenarios where two distinct unknown quantities are linked by two independent linear conditions. A system of two linear equations in two variables ($x$ and $y$) is expressed in standard form as:

$a_1 x + b_1 y = c_1$ $a_2 x + b_2 y = c_2$

Analytical Method 1: The Substitution Method

Ideal when at least one variable in either equation has a coefficient of $+1$ or $-1$:

1. Express one variable explicitly in terms of the other from the simpler equation (e.g., $y = \frac{c_1 - a_1 x}{b_1}$).
2. Substitute this resulting expression into the second equation, creating a single-variable equation.
3. Solve for the remaining variable.
4. Back-substitute the numerical value into your expression from Step 1 to determine the second variable.

Analytical Method 2: The Elimination Method

Ideal when variable coefficients are arbitrary integers:

1. Align both equations in standard form ($ax + by = c$).
2. Multiply one or both equations by non-zero scalar constants so that the coefficients of one variable become equal in magnitude (either identical or opposite in sign).
3. Add the equations (if coefficients have opposite signs) or subtract them (if coefficients have identical signs) to eliminate that variable.
4. Solve the resulting single-variable equation, then back-substitute to find the partner variable.

Worked Example 2: Tactical Logistics Procurement System

A Ghana Armed Forces battalion logistics officer procures two types of specialized field gear: tactical aerial reconnaissance drones ($d$) and secure satellite radios ($r$). The procurement records indicate:

• Order 1: 3 reconnaissance drones and 4 satellite radios cost GH¢29,000.
• Order 2: 5 reconnaissance drones and 2 satellite radios cost GH¢25,000.

Determine the individual unit cost of a reconnaissance drone and a satellite radio.

Step 1: Formulate the simultaneous system.

$\text{Equation (1): } 3d + 4r = 29000$ $\text{Equation (2): } 5d + 2r = 25000$

Step 2: Apply the Elimination Method. Notice that the coefficient of $r$ in Equation (1) is 4, and in Equation (2) it is 2. Multiply Equation (2) by 2 to equate the coefficients of $r$:

$2 \times (5d + 2r) = 2 \times 25000 \implies 10d + 4r = 50000 \quad \text{--- Equation (3)}$

Step 3: Subtract Equation (1) from Equation (3) to eliminate $r$.

$(10d + 4r) - (3d + 4r) = 50000 - 29000$

$7d = 21000$

$d = \frac{21000}{7} = 3000$

The unit cost of a tactical drone is GH¢3,000.

Step 4: Back-substitute $d = 3000$ into Equation (1) to determine $r$.

$3(3000) + 4r = 29000$

$9000 + 4r = 29000$

$4r = 29000 - 9000 = 20000$

$r = \frac{20000}{4} = 5000$

The unit cost of a satellite radio is GH¢5,000.

Step 5: Verification. Check in Equation (2): $5(3000) + 2(5000) = 15000 + 10000 = 25000$. Confirmed.

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Graphical Interpretation & System Classification

Every linear equation $ax + by = c$ corresponds geometrically to a straight line in the Cartesian coordinate plane with slope $m = -\frac{a}{b}$ and y-intercept $k = \frac{c}{b}$. The solution to a simultaneous system represents the geometric intersection point $(x, y)$ of the two lines.

The Determinant Quick-Test

For a system $a_1 x + b_1 y = c_1$ and $a_2 x + b_2 y = c_2$, compute the determinant of coefficients:

$D = a_1 b_2 - a_2 b_1$

• If $D \neq 0$, the system is consistent and independent with a unique solution.
• If $D = 0$, the lines are parallel. If the constant ratios match, the system has infinitely many solutions; if they do not match, the system has no solution.

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Linear Inequalities & Compound Systems

An inequality asserts that one mathematical quantity is greater than ($>$), greater than or equal to ($\ge$), less than ($<$), or less than or equal to ($\le$) another. While a linear equation isolates discrete points, a linear inequality defines a continuous set or interval of numbers on the real line.

The Cardinal Inversion Rule

All standard algebraic operations apply to inequalities with one critical exception:

Mathematical Demonstration: We know that $3 < 7$ is true. If we multiply both sides by $-2$ without reversing the sign, we get $-6 < -14$, which is false ($-6$ is larger than $-14$). Reversing the sign yields $-6 > -14$, which preserves mathematical truth.

Interval Notation & Number Line Conventions

• Strict Inequalities ($<$ or $>$): Indicated on a number line by an open circle ($\circ$) to signify that the boundary value is excluded. In interval notation, boundaries are enclosed by round parentheses: $(a, b)$.
• Inclusive Inequalities ($\le$ or $\ge$): Indicated on a number line by a solid filled dot ($\bullet$) to signify that the boundary value is included. In interval notation, boundaries are enclosed by square brackets: $[a, b]$.

Worked Example 3: Compound Inequality with Sign Reversal

Solve the compound inequality for $x$, express the solution set in interval notation, and define its number line representation:

$-7 \le 5 - 3x < 14$

Step 1: Subtract 5 across all three segments.

$-7 - 5 \le -3x < 14 - 5$

$-12 \le -3x < 9$

Step 2: Divide all three segments by $-3$ and reverse both inequality signs.

$\frac{-12}{-3} \ge x > \frac{9}{-3}$

$4 \ge x > -3$

Step 3: Rewrite in standard ascending orientation.

$-3 < x \le 4$

Solution Set:

• Interval Notation: $(-3, 4]$
• Number Line Representation: An open circle at $-3$, an unbroken line segment extending rightward, terminating in a solid filled dot at $4$.

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Applied Operational Constraints: Payload Distribution

Military decision-making frequently requires maximizing asset distribution within rigid mechanical thresholds.

Worked Example 4: Tactical Airlift Payload Allocation

A Ghana Air Force tactical transport aircraft is assigned to deliver emergency supplies to a forward operating base. The maximum cargo payload capacity is 3,200 kg. The cargo consists of standard infantry ration crates weighing 40 kg each and heavy ammunition cases weighing 75 kg each. Tactical standing orders mandate that the flight must carry a minimum of 20 heavy ammunition cases.

What is the maximum number of infantry ration crates the aircraft can legally transport?

Step 1: Define variables and formulate constraints. Let $r$ be the number of ration crates ($r \in \mathbb{Z}^+$), and $a$ be the number of ammunition cases ($a \in \mathbb{Z}^+$).

• Total weight constraint: $40r + 75a \le 3200$
• Minimum ammunition constraint: $a \ge 20$

Step 2: Substitute the minimum operational constraint to maximize $r$. To maximize the capacity available for ration crates, set ammunition cases to the minimum required threshold ($a = 20$):

$40r + 75(20) \le 3200$

$40r + 1500 \le 3200$

$40r \le 3200 - 1500$

$40r \le 1700$

Step 3: Solve for $r$.

$r \le \frac{1700}{40} = 42.5$

Step 4: Interpret the physical reality constraint. Ration crates cannot be divided into fractional units without destroying packaging integrity. Therefore, we take the floor of $42.5$:

$r_{\max} = 42$

The aircraft can carry a maximum of 42 infantry ration crates.

Step 5: Sanity Check Verification: Total weight: $40(42) + 75(20) = 1680 + 1500 = 3180\text{ kg}$. Margin remaining: $3200 - 3180 = 20\text{ kg}$. Carrying 43 crates would weigh $1720 + 1500 = 3220\text{ kg}$, which violates the 3,200 kg structural limit.