The equation is a mathematical expression that is enclosed between two statements. It contains equals (=) symbol. It says that two things are equal. ‘=’ is an operator that is used only in the equations. Read further to know more about the mathematical sentence, open sentence, equations, and type of equations. You can also find some examples of the equations in the below sections along with solutions for a better understanding of the concepts.

## Equation Definition

A statement of equality of two algebraic expressions that involve one or more variables is called the equation. Every equation has an “=” symbol. The statement or expression at the left side of the “=” symbol is called the left-hand side expression and the expression which is present at the right side of the symbol is called the right-hand side expression.

Example:

4 – x = 10

The set of values of x which satisfy the equation is called the solution set.

### Open Sentence & Mathematical Sentence

The mathematical sentence is a statement whether it is true or false but not both. Some examples are 2 + 3 = 5 which is true, 8 – 5 > 1 which is also true. 8 x 2 <15 a false statement.

Let us take one sentence 3 + x = 18 that may be true or false based on the value of x. So, these type of mathematical sentences having variables become either true or false depending on the literal value is called an open sentence.

### Types of Algebraic Equations

We have different types of equations and they are given along the lines

**1. Linear Equation**: The terms of the linear equations are either a constant or a single variable or a product of both. The general form of the linear equation with two variables is the slope-intercept form. It is given by y = mx + c

Here m is not zero.

m is the slope

c is the y-intercept

The examples of Linear equations with one variable is x + 1 = 8, with two variables is x + y = 1.

**2. Quadratic Equation**: It is a second-order linear equation with two variables. In these equations, any one of the variable exponents must be 2. The general form of quadratic equation is ax² + bx + c = 0, a ≠ 0.

An example of quadratic equation is 5x² + 2y = 25

**3. Radical Equation: **In radical equations, the variable highest exponent is ½ or you can say that the variable is lying inside the square root.

An example of a radical equation is √x – 6 = 30

**4. Exponential Equation: **In this type of equation, the variables are there in place of exponents. By using the exponential equation property, it can be solved.

An example of an exponential equation is 2^x = 56.

**5. Rational Equation: **Rational math equations contain rational expressions.

An example is 8 / x = (x + 7) / 25

### Equation Examples with Solutions

**Example 1.**

Find the solution set for the following open sentences.

(a) x + 5 = 13

(b) x/2 > 5

**Solution:**

(a) x + 5 = 13

If x = 6, then 6 + 5 ≠ 13

If x = 7, then 7 + 5 ≠ 13

If x = 8, then 8 + 5 = 13

Therefore, the solution set for the open sentence x + 5 = 13 is 8.

(b) x/2 > 5

If x = 8, then 8/2 ≯ 5

If x = 9, then 9/2 ≯ 5

If x = 10, then 10/2 ≯ 5

If x = 11, then 11/2 > 5

If x = 12, then 12/2 > 5

Therefore, the solution set for the open sentence x/2 > 5 is 11, 12, 13, . . .

**Example 2.**

Find the solution set for the equations.

(a) 2x + 8 = 18 (b) x + 7 = 14

**Solution:**

(a) 2x + 8 = 18

If x = 3, then 2(3) + 8 = 6 + 8 ≠ 18

If x = 4, then 2(4) + 8 = 8 + 8 ≠ 18

If x = 5, then 2(5) + 8 = 10 + 8 = 18

Therefore, the solution set for the equation 2x + 8 = 18 is 5.

(b) x + 7 = 14

If x = 5, then 5 + 7 ≠ 14

If x = 6, then 6 + 7 ≠ 14

If x = 7, then 7 + 7 = 14

Therefore, the solution set for the equation x + 7 = 14 is 7.

**Example 3.**

Find the values of variables for the following.

(a) 5^x = 125

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

**Solution:**

(a) 5^x = 125

125 can be written as 5 x 5 x 5 = 5³

5^x = 5³

Here the bases are equal so equate the powers.

x = 3.

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

Cross multiply the numerators and denominators

4x = 2(x + 2)

4x = 2x + 4

4x – 2x = 4

2x = 4

If x = 1, then 2(1) ≠ 4

If x = 2, then 2(2) = 4.

Therefore, x = 2.