Guide to 7 Types of Equations: Definitions and Uses

Updated December 19, 2022

Mathematics includes a range of applications that use simple mathematical operations and complex equations. Equations assist people in many industries, such as doctors, astronomers, engineers, statisticians, economists and actuaries, in designing objects, analyzing information, understanding physical relationships between objects and predicting future outcomes. Understanding types of equations and their use in various fields may help you choose a career path in a branch of mathematics.

In this article, we explain what equations are, explore equation terminology and discuss seven common types of equations and their uses.

What is an equation?

An equation is a mathematical statement that displays two expressions as equal. It's a statement with an equal symbol between two mathematical operations equivalent in value, with one on the left side of the symbol and one on the right. If there's no equal symbol in the statement, it's an expression instead of an equation. For example, "2x + 6 = 12" is an equation in which "2x + 6" and "12" are two expressions the equal sign separates.

To find a solution to an equation that contains variables, a mathematician determines the value of the variables, which must remain the same on both sides of the equation. When a specific number replaces the variables, it maintains the equality of an equation, which is the solution. For example, in the equation "5x + 2 = 7x - 22," if you assume x is equal to 12, the solution of the equation is then 12. Other examples of equations are:

7 + 10 = 20 - 3
5 + 20 = 5 x 5
(4 x 2) + 2x − 14 = 0
5x + 2 = 10x - 48

Equation terminology

When discussing equations, engineers, actuaries, and data scientists use terms that include:

Term: A term is a group of numbers, letters and brackets multiplied together, such as "(4 x 2)" or "8x."
Expression: An expression is a term or several terms that you add or subtract, such as "(4 x 2) + 3x - 4."
Variable: A variable is a symbol representing a number that can change in value. For example, in the term "5x," "x" is the variable.
Constant: A constant is a number with a fixed and unchangeable value.
Coefficient: A coefficient is a number in front of a variable, and its function is that it multiplies the variable. For example, in the term "12y," "12" is the coefficient.
Symbol: A symbol is a mark or sign that stands for something else. For example, the plus symbol means to add.
Exponent: An exponent tells you how many times a number multiplies by itself.
Polynomial: A polynomial is an expression in which the exponents of all variables are whole numbers.
Quotient: A quotient is the solution of one number divided by another. For example, the quotient of "200 / 20" is "10."
Solve: When you solve an equation or a problem, you find solutions for it.

7 types of equations

Companies use equations daily to calculate results with different values, which are variables. Consider the net profit formula, which is:

Net profit = Total revenue - (Cost of goods sold + Operating expenses + Taxes)

This formula is a simple equation businesses use each reporting period, and if any of the variables on the left side of this equation change, the solution or result may change. Common equations you may use in careers mathematicians pursue include:

1. Linear equations

A linear equation, known as a one-degree equation, has only one line. This equation may have a few variables, but the highest power of each variable is always one, meaning the variable has no exponents. It also has no square roots. Linear equations have one, two or three variables.

A two-variable linear equation describes a relationship in which the value of one variable depends on the value of the other variable. The graph of a linear equation with two variables will be a straight line. Each term in a linear equation is a constant, a single variable or a product of a constant. Examples of linear equations are:

One variable: 8x - 7 = 0
Two variables: 45x + 5j = 456
Three variables: 6y + 7p - 6t = 68

2. Quadratic equations

A quadratic equation is a second-order equation, which means it contains a minimum of one variable in the equation with an exponent of two. Financial professionals and engineers use quadratic equations to help forecast business profits and plot the course of moving objects, respectively. Cars and clocks would not exist without applying quadratic equations in their design. Examples of quadratic equations include:

ax² + bx + c = 0
5x² - 5y - 35 = 0

An equation like "bx − 6 = 0" isn't a quadratic equation as there are no terms with an exponent of 2. Similarly, terms that contain an exponent of 3 or higher also wouldn't be a quadratic equation.

3. Cubic equations

A cubic equation is a third-order equation. These equations contain a minimum of one term that's cubed, meaning at least one variable in the equation has an exponent of 3. Engineers and scientists use cubic equations to model three-dimensional objects. The cubic function helps identify missing dimensions and explores the results of changes in three-dimensional objects.

ax³+ bx²+ cx + d = 0
x- 7x²+ 4x + 12 = 0

4. Radical equations

Radical equations contain a variable within a square root, and the radical of a number is the same as the root of a number. The maximum exponent on a variable in a radical equation is one-half. Economists and accountants use radical equations to help calculate formulas for inflation, depreciation and interest. Examples of radical equations are:

√(x) + 10 = 26
√27 = √9 x √3

5. Exponential equations

An exponential equation is an equation with exponents in which the exponent or a portion of the exponent is a variable. Everyday applications for exponential equations include compound interest calculations, population growth or decline, the loudness of sound and radioactive decay. Doctors use exponential functions to predict how quickly a disease may spread if antigens multiply.

There are three types of exponential equations:

Equations with the same base number on both sides, such as "4x = 42"
Equations with different base numbers you solve to make the same, such as writing "4x = 16" as "4x = 42"
Equations with different base numbers you may not solve to make, including ones you solve with logarithms, such as "4x = 15"

6. Trigonometric equations

Trigonometric equations measure the relationship between the lengths and angles of the sides of triangles. These equations involve one or more trigonometric ratios of unknown angles, which include sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec) and cosecant (cosec) angles. Solutions of the trigonometric equation are all values that satisfy the equation. Since trigonometric equations can be complex, people often solve them with the help of a calculator.

Civil and mechanical engineers use trigonometric equations to calculate torque and forces on bridges or building girders. Doctors use these equations to understand waves, such as ultraviolet light, radiation or water. Examples of a trigonometric equation are:

tan (x) = 3.2
cos2 (x) + 5 sin (x) = 0

7. Rational equations

A rational equation contains rational expressions, and it's the quotient of two polynomials. These equations help describe relationships between distance, speed and time. Statisticians, engineers, actuaries, astronomers and physicists are some professions that use rational equations. Examples of rational equations include:

x / 4 = (x + 12) / 12
x / 2 = (x + 2) / 4