Algebra Basics: Expressions and Equations

Study Notes

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It involves solving problems using letters and numbers. Algebra builds upon arithmetic, which deals primarily with whole numbers, by introducing symbols called variables to represent unknown values and operations such as addition, subtraction, multiplication, division, and exponentiation.

Algebraic Expressions

An algebraic expression is a mathematical phrase made up of constants, variables, and operators like +, −, ×, ÷, =, and ≠. For example, 5x² - 2x + 7 is an algebraic expression where x represents an unknown value, and the operations are subtraction and addition.

Equations

Equations are statements that two sides are equal. In algebra, equations involve variables and often have multiple solutions. For instance, the equation x + 3 = 7 has one solution because there's only one value of x that makes both sides equal to 7.

Variables

Variables are symbols used to represent unknown values or different quantities in algebra. They can take any numerical value, including positive or negative integers, decimals, fractions, and even zero. An example of a variable is 'x', which could stand for anything from the number of apples you have to your age in years.

Solving Equations

Solving equations means finding the value(s) of the variable that makes the equation true. This process usually involves moving all terms to one side of the equation and setting it equal to zero, then isolating the variable. For example, in the equation x + 3 = 7, we would first move the 3 to the right side by subtracting it from both sides, getting x = 7 - 3 = 4.

Coefficients

Coefficients are fixed factors multiplying the terms of an equation. They can be positive or negative integers, rational numbers, or irrational numbers. For instance, in the quadratic equation ax² + bx + c = 0, a, b, and c are coefficients, and they determine the shape and orientation of the parabola represented by the function f(x).

Description

Explore the fundamentals of algebra, including algebraic expressions, equations, variables, and coefficients. Learn how to manipulate symbols to solve mathematical problems and understand the relationships between different quantities.