# Algebraic Mathematical Equations: Definitions, Types and Examples

## Why are equations important?

The importance of mathematical equations in today’s world may not be immediately apparent. Many of the modern inventions you use on a daily basis, such as computers, televisions, satellites, and GPS, wouldn’t exist without equations. Numerous fields, including economics, medicine, engineering, physics, and computer science, to name a few, depend on mathematical equations. Even some well-known equations have had a significant impact on how you live your life. Some examples include:

## What is an equation?

A mathematical statement known as an equation makes two expressions’ values equal to one another. It is a mathematical statement that says “this equals that,” in other words. The left side appears to be a mathematical expression, the middle appears to be an equal sign, and the right side appears to be a mathematical expression. The right side of the equation frequently has a value of zero.

Here are a few illustrations of both straightforward and complex equations:

3 + 5 = 4 + 4.

dy/dx + x5y = x5y7

20×2 – 17x – 63 = 0

Determine the value that the variables must have in order for the equation to be true when solving an equation with variables. A solution is achieved once the variables’ values are determined and the variables make the equation true. Identity equations are true for all values of the variable. Only certain values of the variable are true for conditional equations.

## Types of algebraic equations

Here are the most common types of algebraic equations:

## Equation terminology

Understanding the terms used to describe and explain mathematical equations is also necessary in order to comprehend mathematical equations. When discussing algebraic equations, the following mathematical terms are frequently used:

## Major algebraic equation categories

There are five main types of algebraic equations, each of which has a different expected input and produces an output that can be interpreted in a different way. Each of the five categories can be distinguished by the variable’s position, the behavior of their graphs, and the kinds of functions and operators employed. A brief description of each of the five algebraic categories is given below:

### Polynomial equations

Equations involving polynomials have polynomial expressions on both sides of the equal sign. Polynomials have variable terms and whole number exponents. Each polynomial equation can be categorized according to the number of terms it contains:

Each polynomial equation can also be categorized according to its degree, which is represented by the expression’s highest exponent number:

For example, this polynomial would be a cubic binomial: x3-5. The polynomial y2 – y – 4 is a quadratic trinomial.

### Exponential equations

Each side of the equal sign in exponential equations contains an exponential expression. Similar to polynomial equations, exponential equations also have a variable term in their exponents. When the independent variable has a positive coefficient, exponential functions are said to exhibit exponential growth; when the independent variable has a negative coefficient, they exhibit exponential decay. Equations for exponential growth can show how diseases spread, how the population is growing, or how compound interest works. Exponential decay equations can be used to illustrate scientific phenomena like radioactive decay.

As an illustration, consider the exponential expression: x = 6(y-8) + 12

### Logarithmic equations

On either side of the equal sign in a logarithmic equation is a logarithmic expression. The inverse of exponential equations is logarithmic equations. The exponent you raise the base to in order to get the number equals the log base of the number. For instance, since 25 is 2 to the 5th power, the log2 of 25 is 5. The transcendental number “e,” also known as a natural logarithm, is the most popular logarithmic base. Many intensity scales, such as the Richter scale for measuring earthquakes and the decibel scale for measuring sound, use logarithms. Since the decibel scale has a log base of 10, a sound’s intensity increases tenfold for every decibel that is added.

For instance, the inverse or logarithmic expression for the exponential expression x = 4y would be x = log4 y.

### Rational equations

On either side of the equal sign in rational equations, there is a rational expression. When an algebraic equation has the form b(x) / d(x), where b(x) and d(x) are both polynomials, it is rational. Asymptotes are points in a graph of a rational equation where the x and y values approach but never reach

A rational equation with a vertical asymptote has an x value that the graph never reaches, and as the x value approaches the asymptote, the y value can either go negative or positive to infinity. A rational equation with a horizontal asymptote has a y value that the graph never reaches, and as the y value approaches the asymptote, the x value can either go negative or positive to infinity.

An illustration of a rational expression would be (y-4) / (y2 – 6y + 3)

### Trigonometric equations

On either side of the equal sign in trigonometric equations is a trigonometric expression. Trigonometric equations use the trigonometric functions tan, cos, sin, cot, csc, and sec to describe the proportion between two sides of a right triangle. The independent variable or input is the angle measure, and the dependent variable or output is the ratio. Trigonometric functions are special because they are periodic, which means that after a certain period of time, their graph repeats.

For instance, the trigonometric formula x = sin y describes the proportion of the hypotenuse of a right triangle to its opposite side, where y is the angle measure.

## Equation examples

Here are some examples of algebraic equations:

### Example 1

The following equation can be resolved in a single step:

x + 5 = 9

x = 9 – 5

x = 4

### Example 2

You can solve the following equation in two steps:

3x + 4 = 16

3x = 12

x = 12/3

x = 4

### Example 3

Here is an equation with multiple steps that involves more than two steps:

4x + 3 = x + 12

4x – x = 12 – 3

3x = 9

x = 9/3

x = 3

### Example 4

Here is a linear equation:

(2x+5) / (x+4) = 1

2x+5 = 1(x+4)

2x+5 = x+4

2x-x = 4-5

x = -1

### Example 5

Here is another linear equation:

x+12 = x2 -2

x2+2x+1 = x2-2

x2-x2+2x+1 = -2

2x+1 = -2

2x+4 = 0

2(x+2) = 0

x+2 = 0

x = -2

√x-7 = 3

(√x-7)2 = 32

x-7 = 9

x = 16

### Example 7

√2x-2 = x-1

(√2x-2)2 = (x-1)2

2x-2 = (x-1)(x-1)

2x-2 = x2-2x+1

0 = x2-4x+3

0 = (x-1)(x-3)

### Example 8

Here is a rational equation:

5/x – 5/6 = 5/3

(6x) 5/x – 5/6 = 5/3 (6x)

30 – 5x = 10X

(+5x) 30 – 5x = 10X (+5x)

30 = 15x

2 = x

### Example 9

Here is a logarithmic equation:

log(2x) = 4

10log(2x) = 104

2x = 104

2x = 10,000

x = 5,000

### Example10

Here is another logarithmic equation:

2+5log3(x-1) = 12

5log3(x-1) = 10

log3(x-1) =2

3log3(x-1) = 32

x-1 = 32

x-1 = 9

x = 10

## FAQ

What are the main types of equations?

The point-slope form, standard form, and slope-intercept form are the three main types of linear equations.

How many types equations are there?

Different Types of Equations
• Linear Equation.