Algebra

What you'll learn

Algebra forms the foundation of US Common Core Math assessments, spanning multiple grade levels and testing your ability to manipulate symbols, solve equations, and represent relationships mathematically. This guide covers expressions, equations, inequalities, functions, linear and quadratic relationships, and systems—all core components tested on Common Core assessments from middle school through high school.

Key terms and definitions

Variable — A symbol (usually a letter) that represents an unknown or changing quantity in an expression or equation.

Expression — A mathematical phrase containing numbers, variables, and operations but no equals sign (e.g., 3x + 5).

Equation — A mathematical statement asserting that two expressions are equal, containing an equals sign (e.g., 3x + 5 = 14).

Coefficient — The numerical factor multiplied by a variable in a term (in 7x², the coefficient is 7).

Like terms — Terms that contain the same variables raised to the same powers, which can be combined through addition or subtraction.

Function — A relationship where each input value corresponds to exactly one output value, often written as f(x).

Linear equation — An equation whose graph is a straight line, written in forms such as y = mx + b (slope-intercept) or Ax + By = C (standard form).

Quadratic equation — A second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0.

Core concepts

Simplifying and evaluating expressions

US Common Core Math requires fluency in manipulating algebraic expressions. The order of operations (PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction) governs all simplification.

Combining like terms:

• Identify terms with identical variable parts
• Add or subtract coefficients only
• Example: 5x² + 3x - 2x² + 7x = (5x² - 2x²) + (3x + 7x) = 3x² + 10x

Distributive property:

• a(b + c) = ab + ac
• Essential for expanding brackets: 3(2x - 5) = 6x - 15
• Required for factoring: 12x + 8 = 4(3x + 2)

Evaluating expressions:

• Substitute given values for variables
• Follow order of operations precisely
• Example: If x = -2, evaluate 3x² - 5x + 1 = 3(-2)² - 5(-2) + 1 = 3(4) + 10 + 1 = 23

Solving linear equations and inequalities

Linear equations form the backbone of Common Core algebra assessments. Master these solution procedures:

One-variable linear equations:

1. Eliminate fractions by multiplying all terms by the LCD
2. Distribute to remove parentheses
3. Combine like terms on each side
4. Isolate variable terms on one side, constants on the other
5. Divide by the coefficient of the variable

Multi-step example process:

• Solve: 2(x - 3) + 5 = 3x - 7
• Distribute: 2x - 6 + 5 = 3x - 7
• Combine like terms: 2x - 1 = 3x - 7
• Subtract 2x: -1 = x - 7
• Add 7: 6 = x

Linear inequalities:

• Follow the same steps as equations
• Critical rule: Reverse the inequality symbol when multiplying or dividing by a negative number
• Express solutions using inequality notation or interval notation
• Example: -3x + 5 ≤ 11 → -3x ≤ 6 → x ≥ -2 (inequality reversed)

Working with linear functions and graphs

Common Core assessments extensively test understanding of linear relationships in multiple representations.

Slope-intercept form (y = mx + b):

• m represents the slope (rate of change)
• b represents the y-intercept (where the line crosses the y-axis)
• Most useful for graphing and identifying key features

Calculating slope:

• From two points: m = (y₂ - y₁)/(x₂ - x₁)
• From a graph: rise/run
• Positive slope: line rises left to right
• Negative slope: line falls left to right
• Zero slope: horizontal line
• Undefined slope: vertical line

Standard form (Ax + By = C):

• A, B, and C are integers
• Useful for quickly finding intercepts
• x-intercept: set y = 0, solve for x
• y-intercept: set x = 0, solve for y

Writing equations from context:

• Identify the rate of change (slope)
• Identify the starting value (y-intercept)
• Example: A gym charges $50 enrollment plus $30 per month → y = 30x + 50

Systems of equations

US Common Core Math tests three solution methods for systems of linear equations:

Graphing method:

• Graph both equations on the same coordinate plane
• Solution is the intersection point
• Used when approximate solutions are acceptable or when technology is available

Substitution method:

1. Solve one equation for one variable
2. Substitute that expression into the other equation
3. Solve for the remaining variable
4. Back-substitute to find the other value

Elimination method:

1. Multiply one or both equations to create opposite coefficients for one variable
2. Add equations to eliminate that variable
3. Solve for the remaining variable
4. Substitute back to find the other value

Example using elimination:

• 3x + 2y = 12
• 5x - 2y = 4
• Add equations: 8x = 16, so x = 2
• Substitute: 3(2) + 2y = 12 → 6 + 2y = 12 → y = 3
• Solution: (2, 3)

System types:

• Consistent independent: One solution (lines intersect once)
• Consistent dependent: Infinite solutions (same line)
• Inconsistent: No solution (parallel lines)

Quadratic equations and functions

Quadratic content appears prominently in high school Common Core assessments.

Standard form: y = ax² + bx + c

• The parabola opens upward if a > 0, downward if a < 0
• Vertex represents maximum or minimum point
• Axis of symmetry: x = -b/(2a)

Factoring to solve:

• Factor the quadratic expression completely
• Set each factor equal to zero
• Solve: x² + 5x + 6 = 0 → (x + 2)(x + 3) = 0 → x = -2 or x = -3

Common factoring patterns:

• Difference of squares: a² - b² = (a + b)(a - b)
• Perfect square trinomials: a² + 2ab + b² = (a + b)²
• General trinomials: factor by grouping or trial-and-error

Quadratic formula:

• For ax² + bx + c = 0: x = [-b ± √(b² - 4ac)]/(2a)
• Use when factoring is difficult or impossible
• Discriminant (b² - 4ac) determines number of real solutions:
  • Positive: two real solutions
  • Zero: one real solution
  • Negative: no real solutions

Exponential functions

Exponential relationships appear in US Common Core high school standards.

General form: y = a(b)ˣ or y = abˣ

• a represents the initial value
• b represents the growth factor (b > 1) or decay factor (0 < b < 1)
• Common base: y = a(1 + r)ˣ where r is the rate of growth/decay

Distinguishing linear vs. exponential:

• Linear functions have constant differences in output values
• Exponential functions have constant ratios in output values
• Example: If y increases by 5 when x increases by 1, linear; if y doubles when x increases by 1, exponential

Rational expressions and equations

Higher-level Common Core assessments include rational expressions.

Simplifying rational expressions:

• Factor numerator and denominator completely
• Cancel common factors
• State restrictions (values that make denominator zero)

Solving rational equations:

1. Identify LCD of all denominators
2. Multiply every term by the LCD
3. Solve resulting polynomial equation
4. Check solutions—reject any that make original denominators zero

Worked examples

Example 1: Multi-step linear equation (Common Core Grade 8)

Solve: (2x - 5)/3 + 4 = (x + 7)/2

Solution:

• Multiply all terms by LCD (6): 6[(2x - 5)/3] + 6(4) = 6[(x + 7)/2]
• Simplify: 2(2x - 5) + 24 = 3(x + 7)
• Distribute: 4x - 10 + 24 = 3x + 21
• Combine like terms: 4x + 14 = 3x + 21
• Subtract 3x: x + 14 = 21
• Subtract 14: x = 7
• Check: (2(7) - 5)/3 + 4 = (14 - 5)/3 + 4 = 9/3 + 4 = 3 + 4 = 7
• And: (7 + 7)/2 = 14/2 = 7 ✓

Example 2: System of equations word problem (Common Core Algebra I)

A movie theater sells adult tickets for $12 and child tickets for $8. On Saturday, the theater sold 340 tickets and collected $3,520. How many of each type of ticket were sold?

Solution:

• Let a = number of adult tickets, c = number of child tickets
• Set up system: a + c = 340 (total tickets)
• 12a + 8c = 3520 (total revenue)
• Use substitution: From equation 1, c = 340 - a
• Substitute into equation 2: 12a + 8(340 - a) = 3520
• Distribute: 12a + 2720 - 8a = 3520
• Combine: 4a + 2720 = 3520
• Subtract 2720: 4a = 800
• Divide by 4: a = 200
• Find c: c = 340 - 200 = 140
• Answer: 200 adult tickets and 140 child tickets
• Check: 200 + 140 = 340 ✓ and 12(200) + 8(140) = 2400 + 1120 = 3520 ✓

Example 3: Quadratic equation solving (Common Core Algebra I)

Solve by factoring: 2x² + 7x - 15 = 0

Solution:

• Factor: Find two numbers that multiply to (2)(-15) = -30 and add to 7
• Numbers are 10 and -3
• Rewrite middle term: 2x² + 10x - 3x - 15 = 0
• Factor by grouping: 2x(x + 5) - 3(x + 5) = 0
• Factor out (x + 5): (2x - 3)(x + 5) = 0
• Set each factor to zero: 2x - 3 = 0 or x + 5 = 0
• Solve: x = 3/2 or x = -5
• Check x = 3/2: 2(3/2)² + 7(3/2) - 15 = 2(9/4) + 21/2 - 15 = 9/2 + 21/2 - 30/2 = 0 ✓

Common mistakes and how to avoid them

Mistake: Distributing incorrectly across subtraction — When distributing a negative, students write 3 - 2(x + 5) = 3 - 2x + 10 instead of 3 - 2x - 10. Correction: Distribute the negative to every term inside parentheses: -2(x) + (-2)(5) = -2x - 10.

Mistake: Forgetting to reverse inequality symbols — Students solve -2x > 6 and get x > -3. Correction: When multiplying or dividing by a negative number, reverse the inequality symbol: x < -3.

Mistake: Canceling terms instead of factors — Students write (x + 3)/(x + 5) = 3/5 by canceling x. Correction: Only factors can cancel, not terms. The expression (x + 3)/(x + 5) cannot be simplified further.

Mistake: Dropping solutions in quadratic equations — After factoring (x - 2)(x + 4) = 0, students only report x = 2. Correction: Set each factor equal to zero separately; both x = 2 and x = -4 are solutions.

Mistake: Misidentifying slope from standard form — Students think the coefficient of x is the slope in 3x + 2y = 6. Correction: Convert to slope-intercept form first: 2y = -3x + 6 → y = -3/2·x + 3, so slope is -3/2.

Mistake: Ignoring restricted values in rational equations — Students find x = 2 solving a rational equation but don't check if it makes denominators zero. Correction: Always verify solutions don't create undefined expressions; reject extraneous solutions.

Exam technique for Algebra

Show all algebraic steps clearly — Common Core assessments award partial credit for correct methods even with calculation errors. Write each transformation on a new line, maintaining the equals sign alignment. Examiners cannot award method marks for work not shown.

Check solutions when time permits — Substitute your answer back into the original equation. This catches sign errors and arithmetic mistakes. For system solutions, verify the point satisfies both equations.

Use appropriate representations — Questions often ask you to "represent the situation with an equation" or "sketch the graph." Match your answer format to the command word: write equations algebraically, draw graphs with labeled axes and key points, create tables with at least three values.

State restrictions and domain limitations — When working with rational expressions, real-world contexts, or functions, explicitly note any values that must be excluded or constraints on variables (e.g., "x ≠ 3" or "t ≥ 0 since time cannot be negative").

Quick revision summary

Algebra on US Common Core Math assessments requires mastery of expressions, equations, inequalities, and functions. Key skills include simplifying expressions using distributive property and combining like terms, solving linear and quadratic equations through multiple methods, graphing and writing linear functions in slope-intercept form, solving systems using substitution or elimination, and interpreting algebraic representations in context. Always show complete working, check solutions by substitution, and state domain restrictions. Distinguish between linear (constant differences) and exponential (constant ratios) patterns. Remember to reverse inequality symbols when multiplying or dividing by negative numbers.