Introduction to Mathematical Equations

A Equation is an equality between two algebraic expressions that is satisfied only when the variables within it take on certain values.

In this introductory lesson, we will focus on studying some fundamental concepts related to mathematical equations. We will provide the definition of a solution to an equation and discuss how to verify if a value is a solution.

Starting from the upcoming lessons, we will explore the techniques to solve equations, with a particular emphasis on first-degree equations.

Mathematical Equations

An equality relationship between two mathematical expressions might not be an identity, meaning it's not verified for all real values belonging to .

Consider, for example, the following relationship:

In this case, this equality isn't verified for all values of . For instance, if we set , we have:

Similarly, the equality isn't verified for other values assigned to . For example, it isn't true for , , . We can show that the relationship is only verified for a single value: .

Substituting the number for the variable in the relation, we get:

Thus, the above relationship is not an identity. There's only one value that can be assigned to the variable that makes the equality true.

Finding the values to assign to a variable or variables in an equality so that it's verified constitutes one of the fundamental problems of mathematics and recurs almost always in any scientific or engineering discipline.

An equality for which we seek the values to assign to the variables so that it's verified is called a Mathematical Equation.

Let's consider another example:

This isn't an identity, as there's only one value to assign to for which the equality is true: . Indeed, by substituting the value for , we get:

The variable of the equation above, for which we're trying to find the value, is called the unknown. The value that verifies the equality is called the solution or root of the equation.

Moreover, the numerical terms of both sides of the equation are referred to as known terms.

An equation can also contain more than one unknown. For example:

The above equation has two unknowns, and .

In this chapter of the math guide, we will focus exclusively on equations with only one unknown.

Members of an equation

The two algebraic expressions that compose the equality in an equation are referred to as the members of the equation.

Equation Domain

As we have mentioned, solving an equation means finding that value or those values (depending on how many unknowns the equation contains) that, when substituted for the unknowns, satisfy the equality.

The search for such values takes place within a possible set of values. Let's revisit the above example:

We saw that the solution is . Even if we didn't specify it, we implied that the solution value should be sought within the entire set of real numbers: .

We could also have specified a different search set, for example, the set of positive real numbers: . However, in this way, the above equation would have no solutions since the only solution, , is a negative number.

In general, the set of values to which the solution or solutions of an equation can belong is referred to as the Domain of the equation.

To clarify how an equation might have a solution in one domain but not in another, let's consider the following example:

The solution to this equation is:

Therefore, this equation has a solution in the set of rational numbers but doesn't have a solution if we choose the set of natural numbers as the domain. Indeed, is not a natural number.

From this point on, we assume that, unless specified otherwise, the domain of an equation is the set of real numbers: .

Solutions of an Equation

Solving an equation involves finding all the solutions. The set of values that satisfy the equation's equality is called the set of solutions, often denoted as .

The reverse process to solving an equation is called verification. In other words, it involves verifying whether a particular value is a solution to the equation or not. It simply involves substituting the value in place of the unknown and checking if the two sides of the equation are equal.

To summarize:

Determinate, Indeterminate, and Impossible Equations

The equations seen so far have a well-defined set of solutions, that is, they have a finite number of solutions.

In this case, when the equation has a finite number of solutions, it is called a determinate equation.

However, there are cases where the number of solutions to an equation is infinite.

In such cases, when the number of solutions is infinite, the equation is said to be indeterminate.

Thinking in terms of the number of solutions, there could be one last scenario. Let's clarify with an example:

In cases where the equation has no solutions, it is said to be an impossible equation.

Types of Equations

In general, in mathematics, there are several types of equations. They can be categorized based on their characteristics.

Integer and Fractional Equations

Based on whether the variables in an equation appear in the numerator or denominator of one or more fractions, we can divide the equations into integer equations and fractional equations.

An example of a integer equation can be:

An example of a fractional equation is:

Numerical and Literal Equations

Based on the letters that appear, equations can be grouped into two categories: numerical equations and literal equations.

An example of a numerical equation is:

In it, apart from the variable , only numbers appear.

An example of a literal equation is:

In this case, besides the variable , the letter also appears, representing a parameter of the equation.

What distinguishes a parameter from a variable is that, when solving a literal equation, the value of the variable should be found as a function of the parameter. That is, one must find an expression for the variable in terms of the parameter. Unlike the variable, the parameter can vary within a predefined set or domain. Generally, it's assumed, unless otherwise specified, that the domain in which the parameter can vary is the set of real numbers: .

Returning to the previous example, we can easily determine that the solution to the equation is:

The solution is an algebraic expression in terms of the parameter . At this point, by assigning any value to the parameter , we obtain the corresponding value for the variable .

For example, assigning the value to , we get:

If, on the other hand, we assign the value to , we get:

And so on.

In Summary

In this lesson, we introduced the concept of a mathematical equation. Although we have not yet seen how to solve an equation, this lesson is crucial as it laid out some fundamental concepts.

We saw that an equation is an equality that holds true only for certain values assigned to its literal variables. These values are referred to as solutions of the equation or roots of the equation, while the literal variables are called unknowns.

To solve an equation means to find its solutions. Conversely, verifying that certain values are solutions to the equation means substituting these values in place of the unknowns and checking that the equality holds.

Now, before delving into solving first-degree equations, we first need to study the principles of equivalence. This way, we can derive the rules necessary for solving equations. We will study these principles in the next lesson.