﻿ How to solve first-degree equations. Step-by-step exercises solved.

# How to solve first-degree equations. Step-by-step exercises solved.

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.

## 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:

1. Relocate terms: Pass the terms with x to one member and the numbers to the other member
2. Simplify: Group similar terms
3. 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.

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.

You can never group terms with x and numbers. You can’t operate with them. Don’t forget.
.

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:

1. Remove parenthesis
2. Relocate terms: Pass the terms with x to one member and the numbers to the other member
3. Simplify: Group similar terms
4. 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:

When there is a minus sign in front of a parenthesis, it changes sign to the terms inside the parenthesis

When there is a plus sign in front of a parenthesis, the terms inside the parenthesis stay the same

## 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:

1. Remove denominators
2. Remove parenthesis
3. Relocate terms: Pass the terms with x to one member and the numbers to the other member
4. Simplify: Group similar terms
5. Clear the x

After removing denominators, the first-degree equation already greatly reduces their difficulty.

In the next section, I’m going to explain in detail how to remove the denominators in the first degree equation.

### How to eliminate denominators in a first-degree equation

Let’s start by explaining with an example how to eliminate the denominators.

We have the following equation: The first thing we have to do is to obtain the common denominator of all the denominators of the equation, both from the first member and the second member, since, as with numbers, in order to add and subtract fractions, it is necessary that they have the same denominator.

In this case, I’m going to choose 24, even if it’s not the minimum common multiple, so you’ll see that you can choose any denominator, as long as it’s common.

We leave the denominator prepared and multiply the numerator by its corresponding number to obtain its equivalent fractions.

This number is obtained by dividing the common denominator by the denominator of the original fraction, just as it is done when operating only with numbers.  In each member, we place everything in a single fraction, whose numerator is the sum or subtraction of all numerators.

At this point, we can eliminate the denominator directly. To do this is similar to what happens with transposition of terms. We can multiply both members by the common denominator, which is equivalent to eliminating them:   We are left with a first-degree equation with parentheses.

We continue to remove parentheses and end up solving the first-degree equation :    Whose result is in the form of a fraction that we have had to simplify.

In the end, it is always solved in the same way and that is to repeat the method over and over again. When you’ve solved a few, you won’t have any problems.

To help you catch more practice, I’m going to solve another step-by-step example of first-degree equations with denominators.

## Example of first-degree equations with step-by-step denominators

First-degree equations with denominators are full of “traps” that you have to take into account to solve them well. That’s why we’re going to repeat another example to review the whole procedure once again, step by step and learn it well.

Let’s go with the first example: We get a common denominator and multiply the denominators with the corresponding numbers to transform to their equivalent fractions the original equation:  We group in a single fraction for each member and eliminate denominators:  Eliminate parentheses and finish solving the equation:     In this case the fraction cannot be simplified.

## Example of how to solve first-degree equations with fractions and parentheses

Let’s see how to solve first-degree equations with fractions and parentheses. For example this equation: We first operate within the brackets to remove the parenthesis. To do this, we remember that the number multiplies each of the terms in the parenthesis: We now operate within square brackets grouping similar terms: We remove the square brackets by multiplying by the fraction. Remember that the fractions are multiplied in line. The terms enclosed in the bracket have as denominator 1. We have terms with different denominator. We eliminate denominators, previously obtaining a common denominator and calculating their equivalent fractions: Now we can remove the denominators. We are left with this equation: We continue eliminating the parentheses: If we have a minus sign preceding a fraction whose numerator has more than one term, remember that the minus sign affects all of the numerator’s terms. We see this in more detail in the course.

We no longer have parentheses or denominators. The next step is to bring all terms with x to the first member and terms without x:

to the second member. We group terms: Y we clear the x: Nos has remained as a solution a fraction that cannot be simplified.

These are the steps for solving first-degree equations with denominators and parentheses. The difficulty will depend on each equation.

## How to solve first-degree equations with fractions

I will now explain how first-degree equations are solved with fractions, step by step.

It is called first-degree equations with fractions because most of its terms are fractions, with a single term in the denominator, unlike first-degree equations with denominators, in which numerators have two or more terms.

In a first-degree equation with fractions, we can find a fraction that is multiplying a parenthesis.

This parenthesis prevents us from eliminating the denominators, so the first step should be to remove the parenthesis that is multiplied by a fraction.

These are the steps for solving these types of equations:

1. Remove parentheses multiplied by fractions
2. Remove denominators
3. Remove parenthesis
4. Relocate terms: Pass the terms with x to one member and the numbers to the other member
5. Simplify: Group similar terms
6. Clear the x

Once we no longer have those parentheses, the denominators can be removed normally and we can continue solving the equation without problems.

## Example of how to solve first-degree equations with fractions

Let’s see with an example how first-degree equations are solved with step-by-step fractions:

As we have commented before, we can have fractions that are multiplying by parentheses and that we must eliminate before removing denominators.

We are going to explain it step by step with an example. This is a first-degree equation with fractions. Numerators have only one term: In which we have a fraction that is multiplying to a parenthesis: We must start by deleting that parenthesis because if we don’t, it is not possible to remove denominators.

We remember that, when two fractions are multiplying, they are multiplied online, that is, the numerator by the numerator and the denominator by the denominator.

Following the example, we multiply the parenthesis and then operate on the denominators:  We have already removed that parenthesis. Now we can remove denominators as if it were a first-degree equation with denominators:

The common denominator is 30, which is the minimum common multiple of 2, 4, 5, and 6:

m.c.m (2,3,5,6) = 30    Y now that we have no denominators, we operate on the terms, relocate terms and clear the x as explained in the lesson on how to solve a first-degree equation:    In this case the result cannot be simplify, but it must be simplified whenever possible.

This is the procedure for solving the first degree equations with fractions.

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## 5 tricks to easily solve first-degree equations

As in everyday life there are numerous householdtrucos to solve day-to-day problems, I’m going to teach you some tricks to solve first-degree equations easily, so that you have more skill when doing your first-degree equations and you can save a lot of time performing operations.

But to make better use of these tricks, you need to know how to solve first-degree equations.

Before I begin to tell you some tricks, I have to make it clear to you that in order to do them, you have to be sure that you know what you are doing and why you are doing it. If you have doubts, it’s preferable that you don’t.

Let’s go with them:

### #1 Repeated terms can be crossed out

When a term is repeated exactly the same in the two terms, we can cross out those terms.

For example, in this equation: We see that the -3x is repeated in the two members: Well, when this happens, we can cross them out directly and erase them from the equation. Because when we reorder terms, among them, the result is 0: This is one of the most useful tricks for solving equations, because we eliminate terms before operating with them and simplify the equation.

### #2 Lesser signs affecting the whole limb can be crossed out

As in the previous point, if we have a minus sign that affects the whole member, it can be crossed out.

But it is very important that it affects the whole term, otherwise we would be modifying the original equation.

For example, in this equation we have a minus sign in each member, which affects the whole member: Therefore, we can cross it out and the new members remain positive: When clearing the x, there would be a positive fraction

It can be crossed out because if we pass -3 to the other member dividing, the minus signs when dividing give a positive sign by the rule of signs: It is important to emphasize that it is necessary for the minus sign to affect the whole member in order to use this trick. For example, in this equation it would not be possible to cross out the minus signs: Because the lesser signs only affect one term.

In order for them to affect the whole member, we would need a parenthesis, as in this case: In the next lesson, when we start with the parentheses, you will be able to apply this trick.

### #3 Denominators affecting the whole member can be crossed out

With the denominators, we have the case that the same denominator divides all the first member and all the second member: In this case, we can also cross out the denominators in order to eliminate them: We can do this because if we passed one of the 4 multiplying to the other member, at the end they would cancel each other out as being equivalent to multiplying by 1 to the second member and we would be left with an equation without denominators. Do not forget that it is absolutely necessary that the denominator affects the whole member.

### #4 We can pass the unknown to the other member to make it positive

Another way to remove the minus sign in front of the x. Let’s see it with this example: Instead of passing the -1 that multiplies x, we can treat -x as a term that is subtracting from the first member and pass it adding to the second member: Now, to clear x, we pass 2 to the first member, which is adding and subtracting: At the end, we can exchange the members of place: Not to confuse with passing terms. What we have done is change one member for another, but the sign of the terms does not vary.

### #5 The minus sign in a fraction belongs to the whole fraction

Although this trick is not just about equations, I thought it would be useful to remember it to make it clearer

There are times when the minus sign is not known whether to place it in the numerator or in the denominator. Then know that it doesn’t matter.

Although the minus sign belongs to the numerator or denominator, in the end the fraction is negative and the minus sign can be placed in front of the fraction: And for the moment nothing else. I hope you can use these tricks to solve first-degree equations and you can use them.