Algebra Notes Flashcards

Absolute Value and Operations

• Absolute value signs function like parentheses, indicating the magnitude of a number regardless of its sign.
• Addition and subtraction operations are performed in a left-to-right sequence.
• Multiplication and division also follow a left-to-right order.

Linear Equations and Expressions

• A linear equation features variables to the power of 1.
• Equations have an equal sign; expressions do not. Changes in equations affect both sides, while expressions maintain a constant value.

Simplifying Expressions

• Combine like terms to simplify.
• Identify a common denominator when necessary.
• Factor out common elements to simplify further.
• Cancel common factors when possible.

Solving Systems of Equations

• Systems of equations can be solved through substitution or elimination methods.

Substitution Method

• Rearrange one equation to isolate a variable.
• Substitute this variable into the other equation.
• Solve for the second variable and substitute back to find the first.

Combination/Elimination Method

• Align equations by adding or subtracting to eliminate a variable.
• Adjust coefficients as needed for ease of elimination.
• Solve the resulting equation and substitute back into the original equation for the remaining variable.

Key Strategies for GMAT

• Look for symmetries and align variables when solving three equations.

Absolute Value Equations

• Typically result in two possible solutions.

Solving Absolute Value Equations

• Isolate the expression within absolute value signs.
• If |x| = z (z > 0), consider x = ±a, creating two cases.
• Validate both solutions by substituting back into the original equation.

Powers of Zero and One

• Any number (including 0) raised to any power equals 0.
• Any number (including 1) raised to any power equals 1.

Solving for Specific Values

• The equation x = x² has solutions of either 0 or 1.
• Raising a positive proper fraction to an exponent decreases its value.

Effects of Exponents on Decimals

• Decimals between 0 and 1, when raised to an exponent, also decrease in value.

Compound Bases and Exponents

• A compound base contains variables or remains undivided into prime factors.
• Raising a compound base to an exponent expands each factor’s exponent.

Multiplying and Dividing Bases

• To multiply same base terms, add their exponents (e.g., x² × x³ × x⁴ = x⁹).
• To divide, subtract the exponent of the denominator from that of the numerator (e.g., 5⁶ ÷ 5² = 5⁴).

Adding and Handling Exponents

• Adding same-term bases involves multiplying the count by the base raised to the same exponent (e.g., 3ⁿ + 3ⁿ + 3ⁿ = 3 × 3ⁿ).
• An exponent of zero for any base results in 1.
• A negative exponent indicates a reciprocal (e.g., 3⁻² = 1/3²).

Exponents Raised to Exponents

• When an exponent is raised to another exponent, multiply the exponents (e.g., (z²)⁴ = z⁸).