6.5 Linear Equations, Inequalities, and Systems of Two Equations
Key Takeaways
- Translating verbal phrases into algebraic expressions requires strict attention to subtraction order (e.g., '5 less than x' translates to x - 5, NOT 5 - x).
- Solving multi-step single-variable linear equations involves clearing denominators via the Least Common Denominator (LCD), applying the distributive property, isolating variable terms, and checking for identities (infinite solutions) or contradictions (no solution).
- When multiplying or dividing both sides of a linear inequality by a negative number, the inequality symbol must be reversed (e.g., -3x >= 12 becomes x <= -4).
- Linear equations in two variables can be expressed in slope-intercept (y = mx + b), point-slope (y - y1 = m(x - x1)), or standard form (Ax + By = C); perpendicular lines have negative reciprocal slopes (m1 * m2 = -1).
- A 2x2 system of linear equations yields one unique solution if the lines intersect (slopes differ), no solution if the lines are parallel (identical slopes, different intercepts), and infinitely many solutions if the lines are coincident (identical slopes and intercepts).
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.
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
| Operation | Key Verbal Phrases | Algebraic Translation |
|---|---|---|
| Addition ($+$) | Sum of $x$ and $8$; $x$ increased by $8$; $8$ more than $x$; total of $x$ and $8$ | $x + 8$ |
| Subtraction ($-$) | Difference of $x$ and $5$; $5$ less than $x$; $x$ decreased by $5$; $5$ subtracted from $x$ | $x - 5$ (Order matters!) |
| Subtraction Trap | $x$ less than $5$; subtract $x$ from $5$ | $5 - x$ |
| Multiplication ($\times$) | Product of $6$ and $x$; $6$ times $x$; twice $x$ ($2x$); three-fourths of $x$ | $6x$; $\frac{3}{4}x$ |
| Division ($\div$) | Quotient of $x$ and $4$; ratio of $x$ to $4$; $x$ divided by $4$ | $\frac{x}{4}$ |
| Powers / Roots | Square of $x$; cube of $x$; square root of $x$ | $x^2$; $x^3$; $\sqrt{x}$ |
| Grouping / Equations | Twice the sum of $x$ and $3$; $4$ times the difference of $y$ and $7$ is $18$ | $2(x + 3)$; $4(y - 7) = 18$ |
Simplifying Expressions & Evaluating Polynomials
- Distributive Property: Multiply the external factor across every term inside parentheses:
- Combining Like Terms: Combine terms containing identical variable bases raised to identical powers by summing their numerical coefficients:
- Evaluating Expressions: Replace variables with assigned numerical values enclosed in parentheses to maintain operational hierarchy and prevent sign errors:
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
- Clear Fractions / Decimals: Multiply every term on both sides by the Least Common Denominator (LCD) of all fractions.
- Clear Parentheses: Apply the distributive property.
- Combine Like Terms: Simplify expressions on the left and right sides independently.
- Isolate Variable Terms: Use addition/subtraction properties to move all variable terms to one side and all constant terms to the opposite side.
- 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:
- Step 2 (Simplify and distribute):
- Step 3 (Combine like terms):
- Step 4 (Isolate variable and constant):
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) |
+-------------------+--------------------+------------------------+-----------------------+
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.
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
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:
Linear Forms Reference Table
| Form Name | Algebraic Structure | Parameters & Features |
|---|---|---|
| Slope-Intercept Form | $y = mx + b$ | $m = \text{slope}$; $(0, b) = y\text{-intercept}$ |
| Point-Slope Form | $y - y_1 = m(x - x_1)$ | $m = \text{slope}$; $(x_1, y_1) = \text{given point on line}$ |
| Standard Form | $Ax + By = C$ | $A, B, C \in \mathbb{Z}$, $A \ge 0$; $\text{Slope } m = -\frac{A}{B}$; $y\text{-int} = \left(0, \frac{C}{B}\right)$; $x\text{-int} = \left(\frac{C}{A}, 0\right)$ |
| Horizontal Line | $y = k$ | $\text{Slope } m = 0$; parallel to $x$-axis; no $x$-intercept (unless $y=0$) |
| Vertical Line | $x = h$ | $\text{Slope is Undefined}$; parallel to $y$-axis; no $y$-intercept (unless $x=0$) |
Parallel vs. Perpendicular Lines
- Parallel Lines ($L_1 \parallel L_2$): Have identical slopes and distinct $y$-intercepts:
- Perpendicular Lines ($L_1 \perp L_2$): Have negative reciprocal slopes (their product equals $-1$):
Solving 2x2 Systems of Linear Equations
A system of linear equations comprises two or more equations sharing the same variables:
Method 1: Substitution
Best used when at least one variable has a coefficient of $+1$ or $-1$.
- Isolate one variable in one equation (e.g., $y = 3x - 5$).
- Substitute that expression into the other equation in place of the variable.
- Solve the resulting single-variable equation.
- Back-substitute the numerical value to find the remaining variable.
Method 2: Elimination (Linear Combinations)
Best used when equations are formatted in Standard Form ($Ax + By = C$).
- Multiply one or both equations by non-zero constants so that the coefficients of one variable become exact additive opposites (e.g., $+6y$ and $-6y$).
- Add the equations vertically to eliminate that variable.
- Solve for the remaining variable.
- Back-substitute into either original equation.
Worked Example: Elimination Method
Solve the system:
- Step 1 (Match coefficients): Multiply Equation 2 by 2 to create $-4y$:
- Step 2 (Add equations):
- Step 3 (Back-substitute): Substitute $x = 2$ into Equation 1:
- Solution: The lines intersect at the unique point $(2, 3)$.
What is the solution set for the linear inequality $5 - 3(2x - 4) \ge 4x - 13$?
Which of the following represents the slope-intercept equation of the line that passes through the point $(4, -1)$ and is perpendicular to the line $3x + 2y = 12$?
A collegiate testing center purchased a combination of standard non-programmable calculators at $15 each and graphing calculators at $45 each. The center bought a total of 60 calculators for an aggregate expenditure of $1,500. How many standard calculators and how many graphing calculators were purchased?