Equations

Here we will discuss about equations. An open sentence containing the sign ‘=‘ is called an equation.

Let’s recall how to solve numerical expressions, i.e., 11 + 7 - 3 and algebraic expressions, i.e., 5x - 2 + 10x/2

What is a mathematical sentence?

We already know that the statement 3 + 5 = 8 is true but the statement 7 + 2 = 8 is not true or 2 + 3 < 5 is also not true.

Such mathematical statements which may either be true or false but not both are called mathematical sentences.

What is an open sentence?

The statement 2 + x = 5 may be true or false depending on the value of x.
If x = 1 then 2 + 1 = 5 is false. If x = 3 then 2 + 3 = 5 is true.

Thus, we can say that the mathematical sentence containing the variable becomes either true or false depending upon the value of the variable.

This type of sentence is called an open sentence, thus 2 + x = 5 is an open sentence.

What is an equation?

A statement of equality of two algebraic expressions which involves one or more literals (variables) is called an equation.

3 + x = 7 is an equation.

The set of values of variables which makes the open sentence true is called the solution set.

Note:

Every equation has two sides — L.H.S. (left-hand side) and R.H.S. (right-hand side).

Literals involved in the equation are called variables. These are usually denoted by letters of English alphabet.

An equation may contain any number of variables.

For example:

(i) 5x + 7 = 19 (ii) 2x + 13y = 8 (iii) 5x - 3y + 4z - 14 = 0

Find the solution set for the following open sentences.

(a) x + 4 = 7

(b) x - 3 > 5

(c) x/2 < 10

The solution set for the following open sentences are explained below step-by-step.

(a) x + 4 = 7

Solution:

x + 4 = 7

If x = 0, then 0 + 4 ≠ 7

If x= 1, then 1 + 4 ≠ 7

If x = 2, then 2 + 4 ≠ 7

If x = 3, then 3 + 4 = 7

Therefore, the solution set for the open sentence x + 4 = 7 is 3.

(b) x - 3 > 5

Solution:

x - 3 > 5

If x = 6, then 6 - 3 ≯ 5

If x = 8, then 8 - 3 ≯ 5

If x = 9, then 9 - 3 = 6 > 5

If x = 10, then 10 - 3 > 5

If x = 11, then 11 - 3 > 5

Therefore, the solution set for the open sentence x - 3 > 5 are all the values of the variable greater than 8, i.e., 9, 10, 11, 12...