4.3 Basic Algebra: Linear Equations, Systems of Equations & Inequalities

4.3 Basic Algebra: Linear Equations, Systems of Equations & Inequalities

Operational Imperative: Algebraic reasoning is the mathematical engine of tactical problem-solving. From establishing fire-support coordination lines to computing payload distributions across military air transports, commissioned officers must formulate symbolic equations from verbal problem stems and isolate unknowns with rapid precision. Under the AFPSAT's 72-second-per-item pace, candidates must avoid common algebraic sign errors and employ factoring shortcuts to outpace conventional scratchpad calculation.

---

Order of Operations (PEMDAS/GEMDAS) & Algebraic Sign Discipline

Mathematical expressions must be evaluated according to the universally established operational hierarchy. In military testing, the acronym PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) or GEMDAS (Grouping symbols, Exponents, Multiplication, Division, Addition, Subtraction) governs precedence:

1. [G/P] Grouping Symbols:   Parentheses (), brackets [], braces {}, and fraction bars
2. [E]   Exponents & Roots:  Powers (x²) and radicals (√x), evaluated left-to-right
3. [M/D] Multiply & Divide:  EQUAL PRIORITY — evaluated strictly from LEFT TO RIGHT
4. [A/S] Add & Subtract:     EQUAL PRIORITY — evaluated strictly from LEFT TO RIGHT

The Left-to-Right Priority Trap in Multiplication and Division

A frequent trap on aptitude exams is evaluating multiplication ahead of division simply because the letter "M" precedes "D" in PEMDAS. Multiplication and division possess identical operational rank. They must be executed strictly in order of appearance from left to right: $18 \div 3 \times 4$

• Correct: Evaluate $18 \div 3 = 6$ first, then multiply: $6 \times 4 = 24$.
• Incorrect Trap: Evaluating $3 \times 4 = 12$ first, yielding $18 \div 12 = 1.5$.

Similarly, addition and subtraction hold equal rank and must be resolved from left to right: $15 - 7 + 4 = (15 - 7) + 4 = 8 + 4 = 12 \quad (\text{NOT } 15 - 11 = 4)$

Negative Sign Distribution & Parenthetical Traps

Sign errors account for more than 40% of missed algebra questions on the AFPSAT. Three rules must be strictly observed:

1. Distinction between $-x^2$ and $(-x)^2$:
   • In the expression $-4^2$, the exponent applies only to the base 4: $-(4 \times 4) = -16$.
   • In the expression $(-4)^2$, the exponent applies to the negative integer: $(-4) \times (-4) = +16$.
2. Distributing Negative Signs Across Parentheses: When a subtraction sign or negative coefficient precedes a grouped polynomial, every term inside the parentheses must change its sign: $-(3x - 8) = -3x + 8$ $-4(2x - 5) = -8x + 20 \quad (\text{NOT } -8x - 20)$
3. Fraction Bars Act as Grouping Symbols: In the expression $\frac{6x - 4}{2}$, the division applies to every term in the numerator: $\frac{6x}{2} - \frac{4}{2} = 3x - 2$.

---

Solving Single-Variable Linear Equations: The 4-Step Protocol

A linear equation in one variable is an equation of the form $ax + b = c$ (where $a \ne 0$). Solving the equation means isolating the variable $x$ so that its coefficient is $+1$.

The 4-Step Systematic Protocol

1. Clear Grouping Symbols: Apply the distributive property to eliminate all parentheses, brackets, and negative prefixes.
2. Clear Fractions and Decimals: Multiply every single term on both sides of the equation by the Least Common Denominator (LCD) of all fractions present. If decimals are present, multiply by $10, 100,$ or $1,000$.
3. Collect Variable Terms: Use addition or subtraction to collect all terms containing the unknown variable onto one side of the equation, and move all constant terms to the opposite side.
4. Isolate the Variable: Divide or multiply both sides by the numerical coefficient of the variable.

Worked Linear Equation Demonstration

Solve for $x$: $\frac{3x - 2}{4} - \frac{x - 3}{6} = 2$

• Step 1: Identify LCD. Denominators are 4 and 6. The LCD is 12.
• Step 2: Clear fractions by multiplying every term by 12: $12 \times \left( \frac{3x - 2}{4} \right) - 12 \times \left( \frac{x - 3}{6} \right) = 12 \times 2$ $3(3x - 2) - 2(x - 3) = 24$
• Step 3: Distribute carefully (watching the negative sign on $-2$): $9x - 6 - 2x + 6 = 24$
• Step 4: Combine like terms: $(9x - 2x) + (-6 + 6) = 24$ $7x = 24 \implies x = \frac{24}{7} = 3\frac{3}{7}$

Special Solution Cases

• Conditional Equation: Has exactly one valid root (e.g., $x = 5$).
• Identity (Infinitely Many Solutions): Variables cancel out, resulting in a true equality (e.g., $0 = 0$ or $4 = 4$). Any real number is a valid solution.
• Contradiction (No Solution): Variables cancel out, resulting in a false mathematical statement (e.g., $0 = 7$). No value of $x$ satisfies the equation.

---

Systems of Two Linear Equations in Two Variables

A system of linear equations comprises two equations with two unknown variables (typically $x$ and $y$). On military entrance tests, these systems model logistical resource allocations, vehicle fleet compositions, and troop pay distributions.

The Two Core Solution Methods

1. The Substitution Method (Optimal when a coefficient is $\pm 1$)

• Isolate one variable in whichever equation has a coefficient of $+1$ or $-1$.
• Substitute that algebraic expression into the other equation in place of the isolated variable.
• Solve the resulting single-variable equation.
• Back-substitute the numerical value to find the remaining variable.

2. The Elimination / Addition Method (Optimal when coefficients are $> 1$)

• Multiply one or both equations by non-zero constants so that the coefficients of one variable become additive opposites (e.g., $+6x$ and $-6x$, or $+10y$ and $-10y$).
• Add the two equations together vertically to eliminate that variable.
• Solve the remaining single-variable equation.
• Back-substitute to find the second variable.

Setting Up Systems from Military Scenarios: The "Wheels and Vehicles" Problem

Scenario: A motorized logistics depot houses 35 total vehicles, composed exclusively of 4-wheeled light utility trucks and 6-wheeled transport carriers. An inventory inspection logs a total of 170 road wheels mounted on these 35 vehicles (excluding spare tires). How many 6-wheeled transport carriers are in the depot?

1. Define variables:
   • Let $u = \text{number of 4-wheeled utility trucks}$.
   • Let $t = \text{number of 6-wheeled transport carriers}$.
2. Set up the system of equations: $\begin{cases} u + t = 35 & (\text{Vehicle Count Equation}) \\ 4u + 6t = 170 & (\text{Wheel Count Equation}) \end{cases}$
3. Solve via Elimination: Multiply the vehicle count equation by $-4$ to eliminate $u$: $-4(u + t) = -4(35) \implies -4u - 4t = -140$
4. Add the modified equation to the wheel count equation: -4u - 4t &= -140 \\ + \quad 4u + 6t &= \phantom{-}170 \\ \hline 2t &= 30 \implies t = 15 \end{aligned}$$
5. Determine $u$: $u = 35 - 15 = 20$.
6. Verification: $20$ trucks have $20 \times 4 = 80$ wheels; $15$ carriers have $15 \times 6 = 90$ wheels. Total wheels = $80 + 90 = 170$. There are exactly $15$ six-wheeled carriers.

---

Linear Inequalities & The Cardinal Sign Reversal Rule

An inequality specifies that one mathematical expression is greater than ($>$), less than ($<$), greater than or equal to ($\ge$), or less than or equal to ($\le$) another. Solving linear inequalities follows the exact balancing rules of linear equations, with one vital, non-negotiable exception:

The Cardinal Rule of Inequalities: Whenever you multiply or divide both sides of an inequality by a negative number, you MUST reverse (flip) the inequality symbol ($<$ becomes $>$, $>$ becomes $<$, $\le$ becomes $\ge$, and $\ge$ becomes $\le$).

Why does this rule exist? Consider the true statement: $2 < 5$.

• Multiply both sides by $-1$: $-2$ and $-5$.
• On the number line, $-2$ lies to the right of $-5$. Thus, $-2 > -5$.
• If you fail to reverse the symbol, you produce the false statement $-2 < -5$.

Worked Inequality Problem

Solve for $x$: $7 - 4x \le 27$

• Subtract 7 from both sides: $-4x \le 20$.
• Divide both sides by $-4$ (FLIP the inequality symbol from $\le$ to $\ge$): $x \ge \frac{20}{-4} \implies x \ge -5$

Graphing Inequalities on a Number Line

• Strict Inequalities ($<$ or $>$): Graphed using an open (unfilled) circle at the boundary number, indicating that the boundary itself is excluded from the solution set.
• Inclusive Inequalities ($\le$ or $\ge$): Graphed using a solid (filled) circle at the boundary number, indicating that the boundary itself is included in the solution set.
• Shading Direction: Shade to the left for less than ($<$ or $\le$); shade to the right for greater than ($>$ or $\ge$).

Compound Inequalities

• Conjunction ("AND"): Solves for the intersection of two conditions, written as a double inequality: $a < x \le b$. Values must satisfy both conditions simultaneously. Example: $-3 \le 2x + 1 < 9 \implies -4 \le 2x < 8 \implies -2 \le x < 4$.
• Disjunction ("OR"): Solves for the union of two conditions ($x < -2 \text{ OR } x \ge 5$). The solution set consists of numbers satisfying either condition (graphed as two separate rays pointing away from each other).

---

Factoring Foundations & Quadratic Equation Mechanics

A quadratic equation is a second-degree polynomial equation of the standard form $ax^2 + bx + c = 0$ (where $a \ne 0$). On the AFPSAT, quadratics appear as factoring problems or geometry/area calculations.

Special Factoring Identities

Mastering these three algebraic factoring patterns allows candidates to factor polynomials and perform rapid mental calculations in seconds:

1. Difference of Two Squares: $a^2 - b^2 = (a - b)(a + b)$ Mental Math Application: Evaluate $65^2 - 35^2$ without squaring large numbers manually: $65^2 - 35^2 = (65 - 35)(65 + 35) = 30 \times 100 = 3,000$ Example: Factor $4x^2 - 49 \implies (2x - 7)(2x + 7)$.

2. Perfect Square Trinomials: $(a + b)^2 = a^2 + 2ab + b^2$ $(a - b)^2 = a^2 - 2ab + b^2$ Example: Factor $x^2 + 10x + 25 = (x + 5)^2$. Example: Factor $x^2 - 14x + 49 = (x - 7)^2$.

Factoring Trinomials of the Form $x^2 + bx + c$

To factor $x^2 + bx + c = (x + p)(x + q)$:

1. Find two integers $p$ and $q$ such that their product equals $c$ ($p \times q = c$) and their algebraic sum equals $b$ ($p + q = b$).
2. Rewrite the expression as $(x + p)(x + q)$.

Worked Example: Factor $x^2 - 5x - 24$.

• Find factors of $-24$ that sum to $-5$.
• Factors of $-24$: $(1, -24), (2, -12), (3, -8)$.
• Notice $3 + (-8) = -5$. Thus, $p = 3$ and $q = -8$.
• Factored form: $(x + 3)(x - 8)$.

Solving Quadratics via the Zero Product Property

If $(x - p)(x - q) = 0$, then either $(x - p) = 0$ or $(x - q) = 0$, giving roots $x = p$ and $x = q$.

The Quadratic Formula & Discriminant Analysis

When a quadratic cannot be factored easily by inspection, use the Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ The expression under the radical is the Discriminant ($\Delta = b^2 - 4ac$), which reveals the nature of the roots:

• $\Delta > 0$: Two distinct real roots. (If $\Delta$ is a perfect square, the roots are rational).
• $\Delta = 0$: Exactly one real root (a repeated/double root: $x = -b / [2a]$).
• $\Delta < 0$: No real roots (roots are complex/imaginary).

---

Strategic Test-Taking Heuristics: Back-Solving & Substitution

When complex algebraic manipulation threatens to exceed your 72-second time limit, deploy two strategic testing tactics:

1. Back-Solving (Plugging in Answer Choices)

In single-variable equations where setting up the algebra is convoluted, test the provided answer choices directly into the equation stem:

• Because answer choices are typically arranged in ascending numerical order, always test Choice C (the middle value) first.
• If Choice C produces a result that is too small, eliminate Choice C, A, and B (if the equation is directly proportional); test Choice D.
• If Choice C produces a result that is too large, eliminate Choice C, D, and E; test Choice B.
• Back-solving from the center guarantees reaching the correct option in a maximum of two trials.

2. Number Substitution for Abstract Algebraic Expressions

When an AFPSAT question asks which expression is equivalent to an abstract algebraic identity (e.g., "Which expression is equivalent to $\frac{x^2 - 9}{x + 3}$?"):

• Substitute a small, convenient integer for the variable (e.g., let $x = 2$ or $x = 4$).
• Avoid using 0 or 1, as they produce identical values across exponents ($0^1 = 0^2$, $1^1 = 1^2$) and can lead to division by zero.
• Evaluate the question stem with your chosen integer to establish a target value.
• Substitute the same integer into the four answer choices; the option yielding the target value is the correct answer.

---

Master Summary of Algebraic Laws, Sign Rules & Traps

The following reference table summarizes the essential laws, sign conventions, and common traps tested in the algebraic section of the AFPSAT: