Let’s now see what a **first-degree equation** is and **how to solve first-degree equations** of all kinds: with parentheses, with denominators and with parentheses and denominators at the same time, with exercises solved step by step.

Índice de Contenidos

- 1 What is an equation
- 2 Degree of an equation
- 3 First-degree equations
- 4 How to solve first-degree equations
- 5 Examples of simple first-degree equations
- 6 How to solve first-degree equations with parentheses
- 7 Solved exercises of first-degree equations with parentheses
- 8 How to solve first-degree equations with denominators
- 9 Example of first-degree equations with step-by-step denominators
- 10 Example of how to solve first-degree equations with fractions and parentheses
- 11 How to solve first-degree equations with fractions
- 12 Example of how to solve first-degree equations with fractions
- 13 5 tricks to easily solve first-degree equations
- 13.1 #1 Repeated terms can be crossed out
- 13.2 #2 Lesser signs affecting the whole limb can be crossed out
- 13.3 #3 Denominators affecting the whole member can be crossed out
- 13.4 #4 We can pass the unknown to the other member to make it positive
- 13.5 #5 The minus sign in a fraction belongs to the whole fraction

## What is an equation

Before going to see what a first-degree equation is, if you were asked what an equation is, would you know how to answer it? Let’s see it in a very simple way.

An equation is a mathematical equality characterized by having an unknown element, called an incognita.

For example, here we have an equation where the unknown is the x:

In addition, the equal sign plays a very important role in an equation, since at all times we are talking about an equality:

Each side of the equal sign is called a member. The left side is called the first member and the right side the second member.

This explanation is very basic and not completely complete, but it’s good for you to get an initial idea and become familiar with the terms.

## Degree of an equation

We have already seen that it is an equation, but how do we know if it is a first-degree equation?

And by the way, what is the grade of an equation?

The degree of an equation coincides with the highest exponent to which the unknowns are elevated.

For example: what is the degree in the following equation?

In this equation, in the first member we have the incognita raised to 2 and raised to 3. In the second member the incognita is raised to 1. Remember that if the incognita has no exponent it means that it is raised to 1:

Therefore, the above equation is of grade 3, which is equal to the greatest exponent of the unknowns.

The degree of an equation indicates the number of solutions in the equation. Thus, a first-degree equation has a solution, a second-degree equation has two solutions, and so on.

## First-degree equations

Well, we already know what an equation is and we know how to identify its grade. Let’s focus now on the first degree equations.

First-degree equations are those equations where x is only elevated to 1, or in other words, x appears simply.

As a grade one, you have a solution.

In general, any first-degree equation has this form, once simplified:

When they are not simplified, we can find them with parentheses, brackets, denominators and fractions such as these:

All these equations must be previously simplified in order to solve them. We will see it step by step below.

## How to solve first-degree equations

### What is to solve an equation

First of all let’s clarify what it is to solve an equation. To solve an equation is to find the numerical value that x must have for equality to be certain.

To do this, we have to simplify the equation until we leave the x alone in one of the members, which is what is called clearing the x. We will see it continuously in this one and in later lessons

To solve first-degree equations, it is commonly said that you have to clear x but, what does it mean to clear x?

To clear x we have to go through a series of steps to reduce or simplify the equation.

To do this, in addition to taking into account the hierarchy of operations, these practical rules are followed:

### Practical rules for solving first-degree equations

Although it is advisable to know how the transposition of terms, in practice these practical rules apply:

- When a term is ADDING to a member, it passes to the other member RESTANDO.
- When a term is REMAINING in one member, it passes to the other member. SUMANDO.
- When a term is MULTIPLICATING in one member, it passes to the other member DIVIDING the entire member
- When a term is DIVIDING into one member, it passes to the other member MULTIPLICATING the entire member

Terms can be passed from left to right or vice versa.

### Steps to solve equations

To solve the first-degree equations we will go through a series of steps:

- Relocate terms: Pass the terms with x to one member and the numbers to the other member
- Simplify: Group similar terms
- Clear the x

It should be made clear that this is not the only way to solve a first-degree equation. You could, for example, simplify before relocating, but practice will teach you that.

We will start solving very simple equations and we will increase the difficulty little by little so that you understand everything.

## Examples of simple first-degree equations

Let’s look at an example of how simple first-degree equations are solved. If we understand these types of equations perfectly, it will be easier to understand how other more complicated first-degree equations are solved (with parentheses, denominators, powers…).

We will start from the following equation:

We see that it is first-degree because x is elevated to 1, as we have indicated in the definition of the first-degree equations.

We start with the first step

1 – Relocate terms.

Through the transposition of terms, we have to pass the terms that carry x to the first member and the numbers that do not carry x to the second member. Terms that are already in the corresponding member should not be touched.

To begin with, we focus on the terms with x and forget the rest of the equation.

In the original equation we see that we have two terms with x: 4x, which is already in the first member and -2x which is in the second member and must be passed to the first member.

The 4x we leave it as it is and the 2x that is RESTANDO, passes ADDING to the first member.

Now let’s go with the numbers and forget the rest.

In the original equation we had two numbers (terms without x): the 14 which is already in the second member and the +2 which is in the first member and must be passed on to the second member:

Now, we rewrite the first member, with the terms with x already relocated and the 14 that is already in the second member. The only thing we have to do is to pass the 2, which is ADDING and passes HOLDING to the second member:

We have already completed the first step. We have the terms with x in the first member and the numbers in the second member.

We continue with the second step.

2 – Simplify: Group similar terms.

In this step it is necessary to group the similar terms, that is, to operate on the one hand with the terms with x and on the other hand with the terms without x.

.

In the previous step, the equation was as follows:

We first operate with the terms in x. We operate with the numbers in front of the x:

Now we operate with the numbers we have left in the second term:

That is nothing more than adding and subtracting numbers. We write the first member with the term already simplified and the result of operating in the second member:We already have the two simplified members. To finish, we have the last step, which is to clear the x.

3 – Clear x

We have the equation already with the terms in place and simplified. Let’s now clear the x.

This last step is very easy and you already practiced it in the previous lesson, so it shouldn’t be a problem for you.

We have to leave the x completely alone and right now it has a 6 in front:

As it is multiplying x, it passes to the other member dividing:

Y now we only have to make the division:

Y this is the solution to the equation. If the fraction is not exact, it is simplifies and is left as a fraction.

## How to solve first-degree equations with parentheses

To solve first-degree equations with parentheses just add one more step to the procedure we already know of solving first-degree equations:

- Remove parenthesis
- Relocate terms: Pass the terms with x to one member and the numbers to the other member
- Simplify: Group similar terms
- Clear the x

Once we no longer have any parentheses, we can continue solving the first-degree equation in the same way we have done so far.

Let’s see how to execute this new step and then we’ll see examples of how to solve equations with parentheses.

### How to eliminate parentheses in the first degree equations

When there is parentheses in an equation, it means that there is a number ahead that is multiplying the terms within the parentheses.

In front of the parenthesis there may be: a number, a minus sign, or a plus sign.

In all these cases, multiply the number by all the terms in the parenthesis, taking into account the rule of signs .

For example:

It is very important to take into account the signs, especially when the number multiplying the parenthesis is negative. The procedure would be the same.

There may be a minus sign or a plus sign in front of the parenthesis. In that case it is equivalent to multiplying by -1 or by +1, respectively.

You have a more direct rule in both cases:

## Solved exercises of first-degree equations with parentheses

Once you’ve learned to solve first-degree equations with parentheses, you’ll see that all exercises are solved the same.

We are going to solve a first-degree equation exercise with parentheses:

We remove the parentheses by multiplying the number in front by the members within the parentheses:

We are left with a much simpler, first-degree equation, without parentheses, which we can continue solving:

In the end we have a fraction, which we have had to simplify.

Let’s go with another example:

We have this parenthesis:

It is a parenthesis with a minus sign in front. We change the signs of the terms inside and remove the parentheses:

We are left with an equation if parentheses, which we continue to solve as always:

And to finish another example:

In this case we have three parentheses

The first of them has nothing in front of it, so we just take it out. The other two we remove as we already know:

Again, we have removed all parentheses and can continue to solve the equation:

## How to solve first-degree equations with denominators

First-degree equations with denominators are where most mistakes are made, so pay close attention and don’t skip a step.

Now I’m going to explain how to solve the first-degree equations with denominators by following these steps:

[box type=”info”]

- Remove denominators
- Remove parenthesis
- Relocate terms: Pass the terms with x to one member and the numbers to the other member
- Simplify: Group similar terms
- Clear the x