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13.2 Algebraic Expressions, Linear Equations, Systems, and Functions
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1 Chapter 1: The CSET Program and the Multiple Subjects Examination
1.1 The CSET Program: Which Tests a California Teacher Actually Needs 1.2 CSET Multiple Subjects: Test Structure, Scoring, and Test-Day Mechanics
2 Chapter 2: Language Structure, Linguistics, and Non-Written Communication
2.1 Linguistics and Language Structure: Phonology, Morphology, Syntax, Semantics, and Pragmatics 2.2 First and Second Language Acquisition and ELD Instruction 2.3 Non-Written Communication, Listening, Speaking, and Media Analysis
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5 Chapter 5: Ancient Civilizations and Medieval World History
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6.1 Indigenous North America, Colonial Settlement, and the American Revolution 6.2 Constitutional Foundations, Federalism, and the American System of Government 6.3 Westward Expansion, Sectional Conflict, the Civil War, and Reconstruction 6.4 Industrialization, the Progressive Era, World Wars, and the Civil Rights Movement
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8 Chapter 8: Social Science Skills: Geography, Civics, Economics, and Source Analysis
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9 Chapter 9: Scientific Practices and Physical Science
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10 Chapter 10: Life Science and Ecology
10.1 Cell Biology, Cellular Organization, and Metabolic Processes 10.2 Genetics, Heredity, Molecular Biology, and Evolution 10.3 Ecosystems, Ecological Interactions, Energy Flow, and Biogeochemical Cycles
11 Chapter 11: Earth, Water, and Space Science
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12 Chapter 12: Number Sense, Operations, and Fractions
12.1 Number Classifications, Place Value, Properties of Operations, and Integers 12.2 Number Theory, Prime Factorization, GCF/LCM, and Exponents 12.3 Fractions, Decimals, Percentages, and Proportional Problem Solving
13 Chapter 13: Algebra, Functions, and Geometry
13.1 Ratios, Rates, Proportions, and Scaling Relationships 13.2 Algebraic Expressions, Linear Equations, Systems, and Functions 13.3 Geometric Properties, Two-Dimensional Polygons, and Three-Dimensional Solids
14 Chapter 14: Coordinate Geometry, Measurement, Statistics, and Mathematical Practices
14.1 Coordinate Geometry, Linear Graphs, and Geometric Transformations 14.2 Measurement Systems, Dimensional Analysis, and Precision 14.3 Descriptive Statistics, Data Displays, and Probability Theory 14.4 Mathematical Practices: Reasoning, Modeling, Estimation, and Justification
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16 Chapter 16: Human Development: Cognition, Intelligence, Social and Physical Growth
16.1 Cognitive and Linguistic Development: Piaget, Vygotsky, and Information Processing 16.2 Perspectives on Intelligence, Play, and Individual Differences 16.3 Social-Emotional and Moral Development: Erikson, Kohlberg, and Attachment Theory 16.4 Physical Development, Ecological Systems, and Mandated Reporting
17 Chapter 17: Visual and Performing Arts: Dance, Music, and Theatre
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18 Chapter 18: Visual Art, Arts Integration, and Constructed-Response Mastery
18.1 Elements of Art, Principles of Design, and Visual Analysis 18.2 Art Media, Techniques, World Art History, and Art Criticism 18.3 California Arts Standards and Cross-Curricular Arts Integration 18.4 CSET Constructed-Response Strategy, Scoring Scale, and Domain-Specific Exemplars
13.2 Algebraic Expressions, Linear Equations, Systems, and Functions
Key Takeaways Explain This Topic AI
Algebraic expressions are formed by combining variables, constants, coefficients, and mathematical operations; simplifying expressions requires applying the distributive property and combining like terms sharing identical variable bases and exponents.
Linear equations in one variable are solved by systematically isolating the unknown variable using inverse operations; solutions are categorized as conditional (unique solution x = c x = c x = c ), identities (infinitely many solutions, 0 = 0 0 = 0 0 = 0 ), or contradictions (no solution, 0 = k 0 = k 0 = k for k ≠ 0 k \ne 0 k = 0 ).
Solving linear inequalities requires reversing the inequality direction symbol ( < , > , ≤ , ≥ <, >, \le, \ge < , > , ≤ , ≥ ) whenever both sides are multiplied or divided by a negative number; solution sets are visualized on number lines using open/closed boundary circles and directional rays.
Systems of two linear equations in two variables are classified geometrically as consistent independent (intersecting lines, 1 unique solution), consistent dependent (coincident lines, infinitely many solutions), or inconsistent (parallel lines, no solution).
A mathematical function f : X → Y f: X \to Y f : X → Y maps each input value in domain X X X to exactly one output value in range Y Y Y (verified via the Vertical Line Test); sequences are discrete functions whose terms are generated by constant addition in arithmetic sequences ( a n = a 1 + ( n − 1 ) d a_n = a_1 + (n-1)d a n = a 1 + ( n − 1 ) d ) or constant multiplication in geometric sequences ( a n = a 1 ⋅ r n − 1 a_n = a_1 \cdot r^{n-1} a n = a 1 ⋅ r n − 1 ).
Last updated: August 2026
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.
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: 7 x 3 − 4 x y + 9 x − 15 7x^3 - 4xy + 9x - 15 7 x 3 − 4 x y + 9 x − 15
Variable: A letter or symbol representing a varying or unspecified numerical quantity ( x , y x, y x , y ).
Term: A single number, variable, or product of numbers and variables separated by addition or subtraction operators ( 7 x 3 7x^3 7 x 3 , − 4 x y -4xy − 4 x y , 9 x 9x 9 x , − 15 -15 − 15 ).
Coefficient: The numerical multiplying factor preceding a variable term (in 7 x 3 7x^3 7 x 3 , the coefficient is 7 7 7 ; in − 4 x y -4xy − 4 x y , the coefficient is − 4 -4 − 4 ).
Constant Term: A term with a fixed numerical value containing no variables ( − 15 -15 − 15 ).
Like Terms: Terms that have the exact same variables raised to the exact same powers (e.g., 5 x 2 y 5x^2y 5 x 2 y and − 8 x 2 y -8x^2y − 8 x 2 y are like terms; 5 x 2 y 5x^2y 5 x 2 y and 5 x y 2 5xy^2 5 x y 2 are unlike terms because the exponents differ).
Combining Like Terms & Simplifying Expressions
Simplifying an expression involves applying the Distributive Property ( a ( b + c ) = a b + a c a(b+c) = ab+ac a ( b + c ) = ab + a c ) and combining like terms by adding their numerical coefficients while retaining identical variable bases and exponents: Expression: 3 ( 2 x 2 − 4 x + 5 ) − 2 ( x 2 − 6 x − 7 ) Distribute: = 6 x 2 − 12 x + 15 − 2 x 2 + 12 x + 14 Group Like Terms: = ( 6 x 2 − 2 x 2 ) + ( − 12 x + 12 x ) + ( 15 + 14 ) Simplified Result: = 4 x 2 + 0 x + 29 = 4 x 2 + 29 \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} Expression: Distribute: Group Like Terms: Simplified Result: 3 ( 2 x 2 − 4 x + 5 ) − 2 ( x 2 − 6 x − 7 ) = 6 x 2 − 12 x + 15 − 2 x 2 + 12 x + 14 = ( 6 x 2 − 2 x 2 ) + ( − 12 x + 12 x ) + ( 15 + 14 ) = 4 x 2 + 0 x + 29 = 4 x 2 + 29
Verbal-to-Algebraic Translation Guide
Translating English language prompts into precise algebraic expressions requires careful attention to word order and operation keywords:
English Verbal Phrase Underlying Operation Algebraic Expression Common Student Translation Error
"Five more than twice a number n n n " Addition / Mult 2 n + 5 2n + 5 2 n + 5 Writing 5 + 2 + n 5 + 2 + n 5 + 2 + n (ignoring multiplication)
"Seven less than three times x x x " Subtraction / Mult 3 x − 7 3x - 7 3 x − 7 Writing 7 − 3 x 7 - 3x 7 − 3 x (reversing the minuend and subtrahend)
"The quotient of y y y and 4 4 4 , decreased by 9 9 9 " Division / Sub y 4 − 9 \frac{y}{4} - 9 4 y − 9 Writing y − 9 4 \frac{y - 9}{4} 4 y − 9 (premature grouping)
"Four times the sum of k k k and 6 6 6 " Grouping / Mult 4 ( k + 6 ) 4(k + 6) 4 ( k + 6 ) Writing 4 k + 6 4k + 6 4 k + 6 (failing to distribute across the sum)
"The difference of squares of a a a and b b b " Exponents / Sub a 2 − b 2 a^2 - b^2 a 2 − b 2 Confusing with the square of the difference: ( a − b ) 2 (a - b)^2 ( a − b ) 2
- 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 a x + b = 0 ax + b = 0 a x + b = 0 ( a ≠ 0 a \ne 0 a = 0 ). Solving an equation requires finding the value(s) of the variable that satisfy the equality using inverse operations:
Clear Fractions / Decimals: Multiply every term on both sides by the Least Common Denominator (LCD) or power of 10.
Apply the Distributive Property: Eliminate all parentheses and grouping symbols.
Combine Like Terms on Each Side: Simplify expressions independently on the left-hand and right-hand sides.
Collect Variable Terms on One Side: Add or subtract variable terms so that the variable appears on only one side.
Isolate the Variable Term: Add or subtract constants to move all numerical values to the opposite side.
Divide by the Coefficient: Multiply or divide by the coefficient of the variable to isolate x x x .
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:
Classification Algebraic Outcome Number of Solutions Geometric Meaning Exemplar Equation
Conditional x = c x = c x = c (unique value) Exactly One Single point of intersection on coordinate axis 3 x + 7 = 22 ⟹ 3 x = 15 ⟹ x = 5 3x + 7 = 22 \implies 3x = 15 \implies x = 5 3 x + 7 = 22 ⟹ 3 x = 15 ⟹ x = 5
Identity 0 = 0 0 = 0 0 = 0 (or c = c c = c c = c ) Infinitely Many True for all real numbers ( x ∈ R x \in \mathbb{R} x ∈ R ) 4 ( x + 2 ) = 4 x + 8 ⟹ 4 x + 8 = 4 x + 8 ⟹ 0 = 0 4(x + 2) = 4x + 8 \implies 4x + 8 = 4x + 8 \implies 0 = 0 4 ( x + 2 ) = 4 x + 8 ⟹ 4 x + 8 = 4 x + 8 ⟹ 0 = 0
Contradiction / Inconsistent 0 = k 0 = k 0 = k ( k ≠ 0 k \ne 0 k = 0 ) No Solution ( ∅ \emptyset ∅ ) False statement; no value of x x x makes equation true 2 x + 9 = 2 x − 3 ⟹ 9 = − 3 2x + 9 = 2x - 3 \implies 9 = -3 2 x + 9 = 2 x − 3 ⟹ 9 = − 3 (False! ∅ \emptyset ∅ )
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 . If a < b and c < 0 , then a ⋅ c > b ⋅ c and a c > b c \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} If a < b and c < 0 , then a ⋅ c > b ⋅ c and c a > c b
Conceptual Justification: On a number line, multiplying by a negative number reflects points across the origin ( 0 0 0 ). Since 2 < 5 2 < 5 2 < 5 , reflecting them yields − 2 > − 5 -2 > -5 − 2 > − 5 .
Step-by-Step Worked Example: Solve − 4 ( 3 x − 5 ) ≥ 2 x + 34 -4(3x - 5) \ge 2x + 34 − 4 ( 3 x − 5 ) ≥ 2 x + 34 Distribute: − 12 x + 20 ≥ 2 x + 34 Subtract 2 x : − 14 x + 20 ≥ 34 Subtract 20 : − 14 x ≥ 14 Divide by − 14 (REVERSE SIGN): x ≤ 14 − 14 ⟹ x ≤ − 1 \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} Distribute: Subtract 2 x : Subtract 20 : Divide by − 14 (REVERSE SIGN): − 12 x + 20 ≥ 2 x + 34 − 14 x + 20 ≥ 34 − 14 x ≥ 14 x ≤ − 14 14 ⟹ x ≤ − 1
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 ≤ ).
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: { a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2 \begin{cases}
- a_1 x + b_1 y = c_1 \\
- a_2 x + b_2 y = c_2
- \end{cases} { a 1 x + b 1 y = c 1 a 2 x + b 2 y = c 2
- Geometric and Algebraic Classification Matrix
System Type Consistency & Dependency Slopes & Intercepts Graph Appearance Number of Solutions
Consistent Independent Consistent (has solution), Independent Different slopes: m 1 ≠ m 2 m_1 \ne m_2 m 1 = m 2 Two lines intersecting at a single point ( x , y ) (x, y) ( x , y ) Exactly 1 Unique Solution
Consistent Dependent Consistent (has solution), Dependent Identical slopes and identical y y y -intercepts: m 1 = m 2 , b 1 = b 2 m_1 = m_2, b_1 = b_2 m 1 = m 2 , b 1 = b 2 Two coincident lines (the exact same line graphed twice) Infinitely Many Solutions
Inconsistent Inconsistent (no solution) Identical slopes, different y y y -intercepts: m 1 = m 2 , b 1 ≠ b 2 m_1 = m_2, b_1 \ne b_2 m 1 = m 2 , b 1 = b 2 Two parallel lines that never intersect No Solution ( ∅ \emptyset ∅ )
Methods for Solving Linear Systems
Method of Substitution: Best when one variable has a coefficient of 1 1 1 or − 1 -1 − 1 .
Procedure: Solve one equation for a single variable (e.g., y = 2 x − 3 y = 2x - 3 y = 2 x − 3 ), substitute that expression into the second equation, solve for the remaining variable, and back-substitute to find the second coordinate.
Method of Elimination (Linear Combination): Best when equations are written in standard form ( A x + B y = C Ax + By = C A x + B y = C ).
Procedure: Multiply one or both equations by non-zero constants so that the coefficients of one variable become additive opposites ( + k +k + k and − k -k − k ). Add the equations together to eliminate that variable, solve for the remaining variable, and back-substitute.
- 4. Functions, Relations, and Sequence Formulations
- The Mathematical Definition of a Function
A relation is any set of ordered pairs ( x , y ) (x, y) ( x , y ) . A function f f f from set X X X to set Y Y Y is a specialized relation with a strict uniqueness rule:
Function Definition: A relation in which each input element x x x in the domain is paired with EXACTLY ONE output element y y y in the range .
Domain: The set of all allowable independent input values ( x x x ).
Range: The set of all resulting dependent output values ( y = f ( x ) y = f(x) 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 N = { 1 , 2 , 3 , 4 , … } \mathbb{N} = \{1, 2, 3, 4, \dots\} N = { 1 , 2 , 3 , 4 , … } .
Arithmetic vs. Geometric Sequences Comparison Matrix
Sequence Attribute Arithmetic Sequence Geometric Sequence
Generating Mechanism Repeated addition of a constant difference ( d d d ) Repeated multiplication by a constant ratio ( r r r )
Recursive Definition a n = a n − 1 + d a_n = a_{n-1} + d a n = a n − 1 + d a n = a n − 1 ⋅ r a_n = a_{n-1} \cdot r a n = a n − 1 ⋅ r
Common Parameter Formula Common Difference: d = a n − a n − 1 d = a_n - a_{n-1} d = a n − a n − 1 Common Ratio: r = a n a n − 1 r = \frac{a_n}{a_{n-1}} r = a n − 1 a n ( a n − 1 ≠ 0 a_{n-1} \ne 0 a n − 1 = 0 )
Explicit n n n -th Term Formula a n = a 1 + ( n − 1 ) d a_n = a_1 + (n - 1)d a n = a 1 + ( n − 1 ) d a n = a 1 ⋅ r n − 1 a_n = a_1 \cdot r^{n-1} a n = a 1 ⋅ r n − 1
Finite Series Sum Formula ( S n S_n S n ) S n = n ( a 1 + a n ) 2 = n [ 2 a 1 + ( n − 1 ) d ] 2 S_n = \frac{n(a_1 + a_n)}{2} = \frac{n[2a_1 + (n-1)d]}{2} S n = 2 n ( a 1 + a n ) = 2 n [ 2 a 1 + ( n − 1 ) d ] S n = a 1 ( 1 − r n 1 − r ) ( r ≠ 1 ) S_n = a_1 \left(\frac{1 - r^n}{1 - r}\right) \quad (r \ne 1) S n = a 1 ( 1 − r 1 − r n ) ( r = 1 )
Continuous Function Analog Linear Function: f ( x ) = d x + c f(x) = dx + c f ( x ) = d x + c (constant slope) Exponential Function: f ( x ) = a ⋅ b x f(x) = a \cdot b^x f ( x ) = a ⋅ b x (exponential growth/decay)
Exemplar Sequence 4 , 9 , 14 , 19 , 24 , … 4, 9, 14, 19, 24, \dots 4 , 9 , 14 , 19 , 24 , … ( a 1 = 4 , d = 5 a_1 = 4, d = 5 a 1 = 4 , d = 5 ) 3 , 6 , 12 , 24 , 48 , … 3, 6, 12, 24, 48, \dots 3 , 6 , 12 , 24 , 48 , … ( a 1 = 3 , r = 2 a_1 = 3, r = 2 a 1 = 3 , r = 2 )
5. Quadratic Functions and Equation Foundations
A quadratic equation in standard form is written as: a x 2 + b x + c = 0 ( a ≠ 0 ) ax^2 + bx + c = 0 \quad (a \ne 0) a x 2 + b x + c = 0 ( a = 0 )
Solving Techniques for Quadratic Equations
Factoring via the Zero Product Property: If a x 2 + b x + c = ( p x + q ) ( r x + s ) = 0 ax^2 + bx + c = (px + q)(rx + s) = 0 a x 2 + b x + c = ( p x + q ) ( r x + s ) = 0 , then p x + q = 0 px + q = 0 p x + q = 0 or r x + s = 0 rx + s = 0 r x + s = 0 .
The Quadratic Formula: Derived by completing the square on the general standard form: x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c
The Discriminant ( Δ = b 2 − 4 a c \Delta = b^2 - 4ac Δ = b 2 − 4 a c )
The expression under the radical, Δ = b 2 − 4 a c \Delta = b^2 - 4ac Δ = b 2 − 4 a c , dictates the nature and number of real roots:
Δ > 0 \Delta > 0 Δ > 0 (Positive): Two distinct real solutions (parabola intersects x x x -axis twice).
If Δ \Delta Δ is a perfect square, solutions are rational; if not, solutions are irrational conjugate pairs.
Δ = 0 \Delta = 0 Δ = 0 (Zero): Exactly one repeated real solution (parabola vertex touches x x x -axis at a single tangent point).
Δ < 0 \Delta < 0 Δ < 0 (Negative): No real solutions (two complex/imaginary solutions a ± b i a \pm bi a ± bi ; parabola does not intersect x x x -axis).
Parabolic Geometry of Quadratic Functions ( f ( x ) = a x 2 + b x + c f(x) = ax^2 + bx + c f ( x ) = a x 2 + b x + c )
Orientation: Opens upward if a > 0 a > 0 a > 0 (vertex is a global minimum); opens downward if a < 0 a < 0 a < 0 (vertex is a global maximum).
Axis of Symmetry: Vertical line passing through the vertex: x = − b 2 a x = -\frac{b}{2a} x = − 2 a b .
Vertex Coordinates: ( − b 2 a , f ( − b 2 a ) ) \left(-\frac{b}{2a}, \; f\left(-\frac{b}{2a}\right)\right) ( − 2 a b , f ( − 2 a b ) ) .