6.5 Linear Equations, Inequalities, and Systems of Two Equations

Algebraic Expressions, Linear Equations, Inequalities & Systems

Core Algebra Engine: Linear algebra represents the largest single content domain on the ACCUPLACER Next-Generation QAS exam. Mastery requires translating contextual word problems into rigorous algebraic expressions, manipulating multi-step equations, navigating the sign-reversal rules of inequalities, analyzing coordinate geometry slopes, and solving systems of linear equations.

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Translating Verbal Phrases into Algebraic Expressions

Translating English statements into mathematical expressions requires mapping lexical syntax to precise arithmetic operators.

Verbal to Algebraic Translation Guide

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Simplifying Expressions & Evaluating Polynomials

1. Distributive Property: Multiply the external factor across every term inside parentheses: $$-4(3x - 5y + 2) = -12x + 20y - 8$$
2. Combining Like Terms: Combine terms containing identical variable bases raised to identical powers by summing their numerical coefficients: $$(5x^2 - 3x + 7) - (2x^2 - 8x - 4) = 5x^2 - 3x + 7 - 2x^2 + 8x + 4 = 3x^2 + 5x + 11$$
3. Evaluating Expressions: Replace variables with assigned numerical values enclosed in parentheses to maintain operational hierarchy and prevent sign errors: $$\text{Evaluate } 3x^2 - 2xy + y^3 \text{ for } x = -2, y = 3:$$ $$3(-2)^2 - 2(-2)(3) + (3)^3 = 3(4) - (-12) + 27 = 12 + 12 + 27 = 51$$

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Solving Multi-Step Single-Variable Linear Equations

A linear equation in one variable can be written in the form $ax + b = 0$ ($a \neq 0$).

The 5-Step Linear Solution Algorithm

1. Clear Fractions / Decimals: Multiply every term on both sides by the Least Common Denominator (LCD) of all fractions.
2. Clear Parentheses: Apply the distributive property.
3. Combine Like Terms: Simplify expressions on the left and right sides independently.
4. Isolate Variable Terms: Use addition/subtraction properties to move all variable terms to one side and all constant terms to the opposite side.
5. Isolate the Variable: Divide or multiply by the variable's coefficient to achieve $1x = k$.

Worked Example: Multi-Step Linear Equation with Fractions

Solve for $x$: $\frac{2x - 1}{3} - \frac{x + 2}{4} = \frac{x - 3}{6} + 1$.

• Step 1 (Find LCD): The LCD of denominators 3, 4, and 6 is 12. Multiply all terms on both sides by 12: $$12 \cdot \left(\frac{2x - 1}{3}\right) - 12 \cdot \left(\frac{x + 2}{4}\right) = 12 \cdot \left(\frac{x - 3}{6}\right) + 12 \cdot (1)$$
• Step 2 (Simplify and distribute): $$4(2x - 1) - 3(x + 2) = 2(x - 3) + 12$$ $$8x - 4 - 3x - 6 = 2x - 6 + 12$$
• Step 3 (Combine like terms): $$5x - 10 = 2x + 6$$
• Step 4 (Isolate variable and constant): $$5x - 2x = 6 + 10 \implies 3x = 16 \implies x = \frac{16}{3}$$

Categorizing Solution Sets: Conditional, Identity, or Contradiction

+-----------------------------------------------------------------------------------------+
|                        TYPES OF LINEAR EQUATION SOLUTIONS                               |
+-------------------+--------------------+------------------------+-----------------------+
| Equation Type     | Final Result       | Solution Set           | Geometric Meaning     |
+-------------------+--------------------+------------------------+-----------------------+
| Conditional       | x = c (e.g., x=4)  | Exactly One Solution   | Single Point of       |
|                   |                    | { c }                  | Intersection          |
+-------------------+--------------------+------------------------+-----------------------+
| Identity          | c = c (e.g., 0=0)  | Infinitely Many        | Coincident Lines      |
|                   | (Always True)      | Solutions (All ℝ)      | (Identical)           |
+-------------------+--------------------+------------------------+-----------------------+
| Contradiction     | a = b (e.g., 0=7)  | No Solution            | Parallel Lines        |
|                   | (Always False)     | Empty Set (∅)          | (Never Intersect)     |
+-------------------+--------------------+------------------------+-----------------------+

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Solving & Graphing Linear Inequalities

Linear inequalities use comparison relations: $<, \le, >, \ge$. Solving inequalities mirrors linear equations, with one crucial exception:

The Cardinal Rule of Inequalities: When you multiply or divide both sides of an inequality by a negative number, you MUST reverse (flip) the direction of the inequality sign.

$$-4x \le 20 \implies \frac{-4x}{-4} \ge \frac{20}{-4} \implies x \ge -5$$

Interval Notation & Number Line Graphing

• Strict Inequalities ($<$ or $>$): Graphed with an open circle ($\circ$); denoted in interval notation with parentheses ( ).
• Inclusive Inequalities ($\le$ or $\ge$): Graphed with a closed solid circle ($\bullet$); denoted in interval notation with brackets [ ].

  Inequality: x > -2        Graph: ───(===•===================>   Interval: (-2, ∞)
                                     -2   0

  Inequality: x ≤ 3         Graph: <=======•===]───────────────   Interval: (-∞, 3]
                                           0   3

  Compound: -1 ≤ x < 4      Graph: ───────[====•=======)───────   Interval: [-1, 4)
                                         -1    0       4

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Linear Equations in Two Variables & Coordinate Geometry

The slope $m$ of a non-vertical line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is defined as:

$$m = \frac{\text{Rise}}{\text{Run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \quad (x_2 \neq x_1)$$

Linear Forms Reference Table

Parallel vs. Perpendicular Lines

• Parallel Lines ($L_1 \parallel L_2$): Have identical slopes and distinct $y$-intercepts: $$m_1 = m_2 \quad \text{and} \quad b_1 \neq b_2$$
• Perpendicular Lines ($L_1 \perp L_2$): Have negative reciprocal slopes (their product equals $-1$): $$m_1 \cdot m_2 = -1 \iff m_2 = -\frac{1}{m_1} \quad (\text{e.g., if } m_1 = \frac{3}{4}, \text{ then } m_2 = -\frac{4}{3})$$

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Solving 2x2 Systems of Linear Equations

A system of linear equations comprises two or more equations sharing the same variables:

$$\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$$

Method 1: Substitution

Best used when at least one variable has a coefficient of $+1$ or $-1$.

1. Isolate one variable in one equation (e.g., $y = 3x - 5$).
2. Substitute that expression into the other equation in place of the variable.
3. Solve the resulting single-variable equation.
4. Back-substitute the numerical value to find the remaining variable.

Method 2: Elimination (Linear Combinations)

Best used when equations are formatted in Standard Form ($Ax + By = C$).

1. Multiply one or both equations by non-zero constants so that the coefficients of one variable become exact additive opposites (e.g., $+6y$ and $-6y$).
2. Add the equations vertically to eliminate that variable.
3. Solve for the remaining variable.
4. Back-substitute into either original equation.

Worked Example: Elimination Method

Solve the system:

$$\begin{cases} 3x + 4y = 18 \quad \text{--- (Equation 1)} \\ 5x - 2y = 4 \quad \text{--- (Equation 2)} \end{cases}$$

• Step 1 (Match coefficients): Multiply Equation 2 by 2 to create $-4y$: $$2 \cdot (5x - 2y) = 2 \cdot (4) \implies 10x - 4y = 8 \quad \text{--- (Equation 2')}$$
• Step 2 (Add equations): $$\begin{aligned} 3x + 4y &= 18 \\ +\quad (10x - 4y &= 8) \\ \hline 13x + 0y &= 26 \implies 13x = 26 \implies x = 2 \end{aligned}$$
• Step 3 (Back-substitute): Substitute $x = 2$ into Equation 1: $$3(2) + 4y = 18 \implies 6 + 4y = 18 \implies 4y = 12 \implies y = 3$$
• Solution: The lines intersect at the unique point $(2, 3)$.