LESSON 13
EQUATIONS
Finding the Unknown
What is an Equation?
An equation is a mathematical statement that two expressions are equal, connected by an equals sign. Unlike an expression, an equation can be solved — we find the value of the variable that makes both sides balance.
Think of an equation as a scale. Whatever you do to one side, you must do to the other to keep it balanced. This principle — performing the same operation on both sides — is the foundation of solving every equation in algebra.
One-Step Equations
A one-step equation requires a single inverse operation to isolate the variable. If a number is added to the variable, subtract it from both sides. If a number is multiplied by the variable, divide both sides by it. The goal is always to get the variable alone on one side.
Two-Step Equations
A two-step equation requires two inverse operations. Work in reverse order of operations — undo addition or subtraction first, then undo multiplication or division. Always perform the same operation on both sides of the equation at each step.
Equations with Variables on Both Sides
When variables appear on both sides of the equation, collect them on one side first by adding or subtracting a variable term. Then solve the resulting one- or two-step equation. Always simplify each side fully before moving terms across.
Checking Your Solution
Always verify your answer by substituting the solution back into the original equation. If both sides equal the same number, the solution is correct. This step catches arithmetic errors and builds confidence in your work.
Key Facts
The equals sign (=) was invented by Welsh mathematician Robert Recorde in 1557. He chose two parallel lines because 'no two things can be more equal'.
An equation with one variable and no exponents higher than 1 is called a linear equation. Its solution is always a single number.
Inverse operations undo each other: addition undoes subtraction, multiplication undoes division, squaring undoes square roots.
Equations are used to model real-world problems — from calculating a sale price to determining how long a trip will take at a given speed.