Introduction to Algebra

• Algebra involves mathematical statements describing relationships.
• Algebraic formulas link numbers via mathematical operations.
• Variables are letters representing numbers which can have different values based on the calculation.

Symbols

• Symbols in algebraic equations dictate operations.
• Addition (+), subtraction (−), multiplication (×), and division (÷) are common symbols.
• The equals sign (=) indicates equivalent values between numbers or formulas.
• Inequalities are shown by > (greater than), < (less than), ≥ (greater than or equal to), and ≤ (less than or equal to).

Terms

• Algebraic terms consist of numbers and variables.
• Examples of terms: 5, -10, x, 6y, and 2x2.
• Constant terms are terms that can only be a number such as 5 and -10.
• A coefficient is a number multiplying a variable; for example, 6 in 6y.
• An exponent is NOT a coefficient, like the 2 in 2x2.
• Like terms share the same variable raised to the same power, such as 3x and -2x.
• Unlike terms have differing variables or the same variable to different powers, such as 4x and 5y or 2x2 and 3x.

Expressions

• Algebraic expressions link terms via mathematical operations.
• Examples of algebraic expressions: 5 + x, 6y − 2x2, and 3x + 4y − 7.

Polynomials

• Polynomials are algebraic expressions with one or more terms.
• Terms include constants, variables, and exponents, excluding division by a variable.
• Exponents must be whole numbers.
• Examples of polynomials: 5, x, 5 + x, 6y − 2x2, and 3x + 4y − 7.
• 3 / x is not a polynomial because it involves division by a variable.
• 2x1/2 is not a polynomial because it has a non-whole number exponent.

Equations

• Algebraic equations demonstrate the equality between two expressions.
• Equals signs are always included in algebraic equations.
• Examples: 5 + x = 10, 6y − 2x2 = 20, and 3x + 4y − 7 = 0.

Evaluating Expressions

• Expression evaluation involves finding its value where the variable equals a number.
• Substituting the variable's value into the expression is needed to simplify and evaluate.
• For example, to evaluate 3x + 5 when x = 2, 2 is substituted for x.
• The expression then reads (3 × 2) + 5 and simplifies to 11.

Solving Equations

• Solving equations involves finding the variable’s value to make the equation true.
• Isolating the variable on one side achieves this.
• Balancing the equation is achieved by performing the same operation to both sides.

Solving Equations - Addition and Subtraction

• According to the addition and subtraction properties of equality, adding or subtracting the same number from both equation sides does not change the solution.
• To solve x − 3 = 7, adding 3 to both sides gives x − 3 + 3 = 7 + 3, so x = 10.

Solving Equations - Multiplication and Division

• Multiplication and division properties of equality state that multiplying or dividing both sides of an equation by the same number doesn't change the solution.
• To solve 5x = 30, divide both sides by 5, giving 5x / 5 = 30 / 5, so x = 6.

Solving Equations - Distributive Property

• The distributive property is a( b + c) = ab + ac.
• This is valuable when solving equations containing parentheses.
• To solve 2(x + 3) = 10, use the distributive property where 2x + 6 = 10, so 2x = 4, and x = 2.

Solving Equations - Combining Like Terms

• Simplifying an equation requires combining like terms before solving.
• To solve 3x + 2x + 5 = 20, combine 3x and 2x to get 5x + 5 = 20, so 5x = 15, and x = 3.

Linear Equations

• Linear equations are algebraic equations where the variable's highest power is 1.
• Linear equations may contain one or more variables.
• A straight line represents the graph of a linear equation.
• The standard form of a one-variable linear equation: ax + b = 0, with constants a and b and a ≠ 0.

Quadratic Equations

• Quadratic equations are algebraic equations where the variable's highest power is 2.
• The standard form of a quadratic equation is ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
• Solving quadratic equations involves factoring, completing the square, or using the quadratic formula.

Quadratic Equations - Factoring

• Factoring rewrites the quadratic equation as a product of binomials.
• For example, x2 + 5x + 6 = (x + 2)(x + 3).
• If (x + 2)(x + 3) = 0, then either x + 2 = 0 or x + 3 = 0, leading to x = −2 or x = −3.

Quadratic Equations - Quadratic Formula

• The quadratic formula solves any quadratic equation.
• Using the standard form ax2 + bx + c = 0, the quadratic formula is: x = (−b ± √(b2 − 4ac)) / (2a).
• x has two potential solutions from this formula.

Systems of Equations

• A system of equations includes two or more equations sharing the same variables.
• The solution is the set of variable values that satisfies all equations.
• Graphing, substitution, or elimination can solve systems of equations.

Systems of Equations - Graphing

• Graphing plots each equation on the same coordinate plane.
• The intersection point of the graphs represents the system’s solution.
• This is best suited for systems of two equations with two variables.

Systems of Equations - Substitution

• Solving for one variable in an equation and substituting that expression into the other equation constitutes substitution.
• This yields a single-variable equation that can be solved.
• Substitute the found variable value back into an original equation for the other variable's value.

Systems of Equations - Elimination

• Adding or subtracting equations to eliminate a variable is elimination.
• This needs multiplying one or both equations by a constant to ensure coefficients of one variable are opposites.
• The resulting single-variable equation can then be solved.
• Substituting that variable's value back into an original equation reveals the other variable's value.

Inequalities

• Inequalities are math statements comparing expressions using inequality symbols.
• The solution is the variable's values that make the inequality true.
• Solving inequalities follows equation methods, except multiplying/dividing by a negative number reverses the inequality sign.

Functions

• Functions show a relationship between inputs and permissible outputs, where each input links to exactly one output.
• Equations, graphs, or tables can represent a function.
• The input is the independent variable.
• The output is the dependent variable.