What is an equation: definition, solution, examples

In this publication, we will look at what an equation is, as well as what it means to solve it. The theoretical information presented is accompanied by practical examples for better understanding.

Equation definition

The equation is , containing the unknown number to be found.

This number is usually denoted by a small Latin letter (most often – x, y or z) and is called variable equations.

In other words, an equality is an equation only if it contains the letter whose value you want to calculate.

Examples of the simplest equations (one unknown and one arithmetic operation):

• x + 3 = 5
• and – 2 = 12
• z + 10 = 41

In more complex equations, a variable may occur several times, and they may also contain parentheses and more complex mathematical operations. For example:

• 2x + 4 – x = 10
• 3 (y – 2) + 4y = 15
• x² + 5 = 9

Also, there can be several variables in the equation, for example:

• x + 2y = 14
• (2x – y) 2 + 5z = 22

Root of the equation

Let’s say we have an equation 2x + 6 = 16.

It turns into a true equality when x = 5. This value (number) is the root of the equation.

Solve the equation – this means finding its root or roots (depending on the number of variables), or proving that they do not exist.

Usually, the root is written like this: x = 3. If there are several roots, they are simply listed separated by commas, for example: x₁ = 2, x₂ =-5.

Notes:

1. Some equations may not be solvable.

For example: 0 · x = 7. Whatever number we substitute for x, it will not work to get the correct equality. In this case, the response is: “the equation has no roots.”

2. Some equations have an infinite number of roots.

For example: and = and. In this case, the solution is any number, i.e. x ∈ R, x ∈ Z, x ∈ NWhere N, Z и R are natural, integer and real numbers, respectively.

Equivalent Equations

Equations that have the same roots are called tantamount to.

For example: x + 3 = 5 и 2x + 4 = 8. For both equations, the solution is the number two, i.e. x = 2.

Basic equivalent transformations of equations:

1. The transfer of some term from one part of the equations to another with a change in its sign to the opposite.

For example: 3x + 7 = 5 tantamount to 3x + 7 – 5 = 0.

2. Multiplication / division of both parts of the equation by the same number, not equal to zero.

For example: 4x – 7 = 17 tantamount to 8x – 14 = 34.

The equation also does not change if the same number is added/subtracted to both sides.

3. Reduction of similar terms.

For example: 2x + 5x – 6 + 2 = 14 tantamount to 7x – 18 = 0.