Introduction to Algebra

Algebra Fundamentals

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities in formulas and equations
• It is a unifying thread of almost all of mathematics

Algebraic Expressions

• An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y) and operators (like add, subtract, multiply, and divide)
• Example: 3x + 5y - 8 is an algebraic expression
• Expressions can be simplified by combining like terms
• Like terms are terms that contain the same variable raised to the same power; for example, 3x and 5x are like terms, but 3x and 5x² are not

Equations

• An equation is a statement that two expressions are equal
• Equations contain an equals sign (=)
• Example: 3x + 5 = 14 is an equation
• To solve an equation, one must find the value(s) of the variable(s) that make the equation true

Solving Linear Equations

• Linear equations are equations in which the highest power of the variable is 1
• To solve a linear equation, isolate the variable on one side of the equation by performing the same operations on both sides
• The goal is to get the variable by itself on one side of the equals sign
• Common operations include addition, subtraction, multiplication, and division

Solving Linear Equations: Example

• Equation: 2x + 3 = 7
• Subtract 3 from both sides: 2x + 3 - 3 = 7 - 3 which simplifies to 2x = 4
• Divide both sides by 2: (2x)/2 = 4/2 which simplifies to x = 2
• Therefore, the solution to the equation is x = 2

Systems of Linear Equations

• A system of linear equations is a set of two or more linear equations containing the same variables
• The solution to a system of linear equations is the set of values for the variables that satisfy all equations in the system simultaneously
• Systems of equations can be solved using various methods, including substitution, elimination, and graphing

Solving Systems of Equations: Substitution

• Solve one equation for one variable
• Substitute that expression into the other equation
• Solve for the remaining variable
• Substitute the value back into either equation to find the value of the first variable

Solving Systems of Equations: Elimination

• Multiply one or both equations by a constant so that the coefficients of one variable are opposites
• Add the equations together to eliminate one variable
• Solve for the remaining variable
• Substitute the value back into either equation to find the value of the first variable

Inequalities

• An inequality is a statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to)
• Example: 2x + 3 < 7
• Solving inequalities is similar to solving equations, but with one important difference: when multiplying or dividing both sides by a negative number, the inequality sign must be reversed

Quadratic Equations

• A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0
• Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula
• Factoring involves rewriting the quadratic expression as a product of two linear expressions
• The quadratic formula is: x = (-b ± √(b² - 4ac)) / (2a) giving two possible solutions for x

Polynomials

• A polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents
• Example: 3x² - 7x + 5 is a polynomial
• Polynomials can be added, subtracted, multiplied, and divided
• The degree of a polynomial is the highest power of the variable in the polynomial
• Polynomial long division is a method for dividing polynomials, similar to long division with numbers

Factoring Polynomials

• Factoring a polynomial involves rewriting it as a product of simpler expressions
• Common factoring techniques include factoring out the greatest common factor (GCF), difference of squares, and trinomial factoring

Exponents and Radicals

• An exponent indicates how many times a base number is multiplied by itself
• Example: x³ means x * x * x
• Radicals are the opposite of exponents; the nth root of a number x is a number that, when raised to the nth power, equals x
• Example: √9 = 3 because 3² = 9
• Key exponent rules:
  • Product of powers: xᵃ * xᵇ = xᵃ⁺ᵇ
  • Quotient of powers: xᵃ / xᵇ = xᵃ⁻ᵇ
  • Power of a power: (xᵃ)ᵇ = xᵃᵇ
  • Negative exponent: x⁻ᵃ = 1/xᵃ
  • Zero exponent: x⁰ = 1

Rational Expressions

• A rational expression is a fraction where the numerator and denominator are both polynomials
• Rational expressions can be simplified, added, subtracted, multiplied, and divided
• Simplifying a rational expression involves canceling common factors from the numerator and denominator
• When adding or subtracting rational expressions, a common denominator is required

Functions

• A function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output
• The input is called the argument of the function, and the output is called the value of the function
• Functions are often denoted by symbols such as f(x), where x is the input
• The set of all possible inputs is called the domain of the function, and the set of all possible outputs is called the range of the function

Graphing Linear Equations

• Linear equations can be graphed on a coordinate plane
• The graph of a linear equation is a straight line
• To graph a linear equation, find two points that satisfy the equation and draw a line through them
• The slope-intercept form of a linear equation is y = mx + b, where m is the slope and b is the y-intercept

Word Problems

• Algebra is often used to solve word problems
• To solve a word problem, first identify the unknown quantities and assign variables to them
• Then, translate the problem into an algebraic equation or system of equations
• Finally, solve the equation(s) and interpret the solution in the context of the problem