Solving equations with one unknown (variable)

In this publication, we will consider the definition and general form of writing an equation with one unknown, and also provide an algorithm for solving it with practical examples for a better understanding.

Defining and writing an equation

Mathematical expression of the form ax + b = 0 is called an equation with one unknown (variable) or a linear equation. Here:

• a и b – any numbers: a is the coefficient for the unknown, b – free coefficient.
• x – variable. Any letter can be used for designation, but Latin letters are generally accepted. x, y и z.

The equation can be represented in the equivalent form ax = -b. After that, we look at the odds.

• RџSЂRё a ≠ 0 single root x = -b/a.
• RџSЂRё a = 0 the equation will take the form 0 ⋅ x = -b. In this case:
  • if b ≠ 0, there are no roots;
  • if b = 0, the root is any number, because expression 0 ⋅ x = 0 true for any value x.

Algorithm and examples of solving equations with one unknown

Simple options

Consider simple examples for a = 1 and the presence of only one free coefficient.

Sophisticated options

When solving a more complex equation with one variable, it is very often necessary to first simplify it before finding the root. The following methods can be used for this:

• opening brackets;
• transfer of all unknowns to one side of the “equal” sign (usually to the left), and known ones to the other (right, respectively).
• reduction of similar members;
• exemption from fractions;
• dividing both parts by the coefficient of the unknown.

Example: solve the equation (2x + 6) ⋅ 3 – 3x = 2 + x.

Solution

1. Expanding the brackets:

   6x + 18 – 3x = 2 + x.

2. We transfer all the unknowns to the left, and the known ones to the right (do not forget to change the sign to the opposite when transferring):

   6x – 3x – x = 2 – 18.

3. We carry out the reduction of similar members:

   2x = -16.

4. We divide both parts of the equation by the number 2 (the coefficient of the unknown):

   x = -8.