How to solve first degree equations: the rules, the complete explanation and the solved exercises. Definition and examples of equations.
If you need a clear and simple explanation of first degree equations for your math test , here you will find an in-depth study of the types of first degree equations, with formulas and solved exercises . In short, everything you need to understand the theory of equations and apply it to the best of your ability.
First degree equations: definition
Let’s start with the definitions:
- Expression
A set of numbers and letters connected by operations to be performed on them constitute an expression. The letters that appear in an expression can have two different meanings: they can be constants (generally indicated by the first letters of the alphabet: a,b,c,…), or variables (and indicated by the last letters of the alphabet: x,y,z). - Identity
Identity is the equality between two expressions that occurs for any value assigned to the variables contained in them and for which the expressions have meaning. For example, the following equalities are identities:
10x = 10x
4 x (a + b) = 4a + 4b - Equation
An equation is an equality between two expressions that is verified only for particular values (called solutions) assigned to the variables (unknowns) contained in it. For example:
| 3x + 2 | = | 7 |
| 1st member | 2nd member | |
What does it mean to solve a first degree equation?
Solving an equation means determining the set of solutions S, that is, the set of those particular values that, assigned to the variables, satisfy the equation, transforming it into equality. Note that, a priori, given an equation, we do not know if solutions exist.
It could also happen that, by solving it, we discover that it is satisfied for any value of the unknown: in this case, we discover a posteriori that the given equation was in fact an identity .
So, be careful: when we have an equation in front of us (that is, an expression in which one or more unknowns appear) our goal is to solve it, that is, to find values that, substituted for the unknown, give rise to an identity (for example: 7=7). The case in which the equation is shown to be, a posteriori, an identity, that is, verified for each value of the unknown that does not make the operations that appear lose their meaning, is not excluded.
For example:
5x-3=4x+1 is an equation that admits x=4 as a solution: in fact, by substituting this value for x we obtain the identity 17=17.
We will soon see how to solve an equation.
In the meantime, let’s see a first classification of equations based on the existence and number of solutions.
