13.2 Algebraic Expressions, Linear Equations, Systems, and Functions

13.2 Algebraic Expressions, Linear Equations, Systems, and Functions

CSET Focus: Algebraic thinking on CSET Multiple Subjects Subtest II spans the structural transition from generalized arithmetic to formal function analysis. Candidates must be fluent in simplifying algebraic expressions, solving multi-step equations and inequalities (including the negative multiplication sign reversal rule), solving and classifying systems of linear equations (consistent vs. inconsistent), understanding function notation and domain/range mappings, working with arithmetic and geometric sequences, and solving foundational quadratic equations via factoring and the quadratic formula.

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1. Algebraic Expressions, Terms, and Verbal Translations

An algebraic expression is a mathematical phrase composed of numbers (constants), letters representing unknown quantities (variables), and arithmetic operation symbols ($+, -, \times, \div$). Unlike equations, expressions contain no relational equals sign ($=$) and cannot be "solved"—they can only be simplified or evaluated.

Structural Anatomy of an Expression

Consider the polynomial expression: $7x^3 - 4xy + 9x - 15$

• Variable: A letter or symbol representing a varying or unspecified numerical quantity ($x, y$).
• Term: A single number, variable, or product of numbers and variables separated by addition or subtraction operators ($7x^3$, $-4xy$, $9x$, $-15$).
• Coefficient: The numerical multiplying factor preceding a variable term (in $7x^3$, the coefficient is $7$; in $-4xy$, the coefficient is $-4$).
• Constant Term: A term with a fixed numerical value containing no variables ($-15$).
• Like Terms: Terms that have the exact same variables raised to the exact same powers (e.g., $5x^2y$ and $-8x^2y$ are like terms; $5x^2y$ and $5xy^2$ are unlike terms because the exponents differ).

Combining Like Terms & Simplifying Expressions

Simplifying an expression involves applying the Distributive Property ($a(b+c) = ab+ac$) and combining like terms by adding their numerical coefficients while retaining identical variable bases and exponents:

$$\begin{aligned} \text{Expression:} \quad & 3(2x^2 - 4x + 5) - 2(x^2 - 6x - 7) \\ \text{Distribute:} \quad & = 6x^2 - 12x + 15 - 2x^2 + 12x + 14 \\ \text{Group Like Terms:} \quad & = (6x^2 - 2x^2) + (-12x + 12x) + (15 + 14) \\ \text{Simplified Result:} \quad & = 4x^2 + 0x + 29 = 4x^2 + 29 \end{aligned}$$

Verbal-to-Algebraic Translation Guide

Translating English language prompts into precise algebraic expressions requires careful attention to word order and operation keywords:

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2. Linear Equations and Multi-Step Inequalities in One Variable

The Systematic Multi-Step Equation Protocol

A linear equation in one variable can be written in the standard form $ax + b = 0$ ($a \ne 0$). Solving an equation requires finding the value(s) of the variable that satisfy the equality using inverse operations:

1. Clear Fractions / Decimals: Multiply every term on both sides by the Least Common Denominator (LCD) or power of 10.
2. Apply the Distributive Property: Eliminate all parentheses and grouping symbols.
3. Combine Like Terms on Each Side: Simplify expressions independently on the left-hand and right-hand sides.
4. Collect Variable Terms on One Side: Add or subtract variable terms so that the variable appears on only one side.
5. Isolate the Variable Term: Add or subtract constants to move all numerical values to the opposite side.
6. Divide by the Coefficient: Multiply or divide by the coefficient of the variable to isolate $x$.
7. Verify Solution: Substitute the resulting value back into the original unsimplified equation.

Classification of Linear Equation Solutions

When simplifying a linear equation, three mutually exclusive outcomes are possible:

Linear Inequalities and the Sign Reversal Theorem

A linear inequality asserts that one algebraic expression is greater than, less than, greater than or equal to, or less than or equal to another ($<, >, \le, \ge$).

The Fundamental Negative Multiplication/Division Rule: Whenever you multiply or divide both sides of an inequality by a negative real number, you MUST reverse the direction of the inequality symbol.

$$\text{If } a < b \text{ and } c < 0, \quad \text{then } a \cdot c > b \cdot c \quad \text{and} \quad \frac{a}{c} > \frac{b}{c}$$

• Conceptual Justification: On a number line, multiplying by a negative number reflects points across the origin ($0$). Since $2 < 5$, reflecting them yields $-2 > -5$.
• Step-by-Step Worked Example: Solve $-4(3x - 5) \ge 2x + 34$

$$\begin{aligned} \text{Distribute:} \quad & -12x + 20 \ge 2x + 34 \\ \text{Subtract } 2x: \quad & -14x + 20 \ge 34 \\ \text{Subtract } 20: \quad & -14x \ge 14 \\ \text{Divide by } -14 \text{ (REVERSE SIGN):} \quad & x \le \frac{14}{-14} \implies x \le -1 \end{aligned}$$

Number Line Graphing Conventions

• Open Circle ($\circ$): Used for strict inequalities ($<$ or $>$) to indicate that the boundary endpoint is excluded.
• Closed Circle ($\bullet$): Used for inclusive inequalities ($\le$ or $\ge$) to indicate that the boundary endpoint is included.
• Directional Ray: Shaded to the right for greater values ($>$ or $\ge$); shaded to the left for lesser values ($<$ or $\le$).

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3. Systems of Linear Equations in Two Variables

A system of linear equations consists of two or more linear equations containing the same set of variables:

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

Geometric and Algebraic Classification Matrix

Methods for Solving Linear Systems

1. Method of Substitution: Best when one variable has a coefficient of $1$ or $-1$.
   • Procedure: Solve one equation for a single variable (e.g., $y = 2x - 3$), substitute that expression into the second equation, solve for the remaining variable, and back-substitute to find the second coordinate.
2. Method of Elimination (Linear Combination): Best when equations are written in standard form ($Ax + By = C$).
   • Procedure: Multiply one or both equations by non-zero constants so that the coefficients of one variable become additive opposites ($+k$ and $-k$). Add the equations together to eliminate that variable, solve for the remaining variable, and back-substitute.

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4. Functions, Relations, and Sequence Formulations

The Mathematical Definition of a Function

A relation is any set of ordered pairs $(x, y)$. A function $f$ from set $X$ to set $Y$ is a specialized relation with a strict uniqueness rule:

Function Definition: A relation in which each input element $x$ in the domain is paired with EXACTLY ONE output element $y$ in the range.

• Domain: The set of all allowable independent input values ($x$).
• Range: The set of all resulting dependent output values ($y = f(x)$).
• Vertical Line Test: A curve in the Cartesian coordinate plane represents a function if and only if no vertical line intersects the curve at more than one point.

  FUNCTION (Passes Vertical Line Test)       NOT A FUNCTION (Fails Test)
         y                                          y
         |   /                                      |    ╭───╮
      ───┼──/─── Vertical line hits              ───┼───│─*─│─── Vertical line hits
         | /     at exactly ONE point               |   │ * │    at TWO points!
         |/                                         |    ╰───╯
  ───────┼───────> x                         ───────┼───────> x

Sequences as Discrete Functions

A sequence is a function whose domain is the set of positive integers $\mathbb{N} = \{1, 2, 3, 4, \dots\}$.

Arithmetic vs. Geometric Sequences Comparison Matrix

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5. Quadratic Functions and Equation Foundations

A quadratic equation in standard form is written as:

$$ax^2 + bx + c = 0 \quad (a \ne 0)$$

Solving Techniques for Quadratic Equations

1. Factoring via the Zero Product Property: If $ax^2 + bx + c = (px + q)(rx + s) = 0$, then $px + q = 0$ or $rx + s = 0$.
2. The Quadratic Formula: Derived by completing the square on the general standard form:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The Discriminant ($\Delta = b^2 - 4ac$)

The expression under the radical, $\Delta = b^2 - 4ac$, dictates the nature and number of real roots:

• $\Delta > 0$ (Positive): Two distinct real solutions (parabola intersects $x$-axis twice).
  • If $\Delta$ is a perfect square, solutions are rational; if not, solutions are irrational conjugate pairs.
• $\Delta = 0$ (Zero): Exactly one repeated real solution (parabola vertex touches $x$-axis at a single tangent point).
• $\Delta < 0$ (Negative): No real solutions (two complex/imaginary solutions $a \pm bi$; parabola does not intersect $x$-axis).

Parabolic Geometry of Quadratic Functions ($f(x) = ax^2 + bx + c$)

• Orientation: Opens upward if $a > 0$ (vertex is a global minimum); opens downward if $a < 0$ (vertex is a global maximum).
• Axis of Symmetry: Vertical line passing through the vertex: $x = -\frac{b}{2a}$.
• Vertex Coordinates: $\left(-\frac{b}{2a}, \; f\left(-\frac{b}{2a}\right)\right)$.