Solving Equations and Inequalities

Foundations of Equations and Properties of Equality

In algebra, an equation is a formal mathematical statement asserting that two expressions are equal by connecting them with an equals sign. Within these mathematical expressions, you will encounter variables, coefficients, and constants. A variable is a letter or symbol that represents a number that may be unknown or may vary. A coefficient is a numerical factor that multiplies a variable, whereas a constant is a fixed numerical value that does not contain a variable. When expressions contain multiple parts, like terms are identified as terms that have identical variable parts raised to the same power. Solving an equation means finding all values of the variable that make the equation true.

To find the value of an unknown variable, equations are manipulated using inverse operations. Inverse operations are operations that reverse the effect of each other. Specifically, addition and subtraction are inverse operations of each other, and multiplication and division are inverse operations of each other. You can think of an equation like a balanced physical scale: whatever action you take to alter one side must be matched on the opposing side to keep the relationship balanced.

Four fundamental properties govern the manipulation of equations. The addition property of equality states that adding the same quantity to both sides of an equation preserves equality. Similarly, the subtraction property of equality states that subtracting the same quantity from both sides of an equation preserves equality. The multiplication property of equality states that multiplying both sides of an equation by the same non-zero quantity preserves equality, and the division property of equality states that dividing both sides of an equation by the same non-zero quantity preserves equality. Applying these properties maintains a true statement throughout the solving process.