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

Next we will explain how to solve first-degree equations with parentheses, with denominators and with parentheses and denominators at the same time, with step-by-step exercises.

The first-degree equations are those equations where the x only appears elevated to 1, or in other words, the x appears simply.

The degree of an equation indicates the number of solutions. In this case, being grade 1, it has only one solution.

The first-degree equations with denominators is where most mistakes are made, so you should pay close attention and not skip any steps.

Solving First Degree Equations

To solve first-degree equations, it is commonly said that the x must be cleared, but what does it mean to clear the x?

Clearing x means leaving it alone in one member of the equation.

And how do you clear the x?

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

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

Practical Rules for Solving Equations

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

  • When a term is ADDING in one member, it passes to the other memer SUBTRACTING.
  • When a term is RESTORED in one member, it passes to the other member ADDING.
  • When a term is MULTIPLYING in one member, it passes to the other member DIVIDING the entire member.
  • When a term is DIVIDING in one member, it passes to the other member MULTIPLYING the entire member

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

Example of how simple first-degree equations are solved

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

We’ll start from the next equation:

ecuaciones de primer grado

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

First of all, you have to pass the terms with x on one side of the equals and the numbers on the other side. Keep in mind that what you are adding in one member happens by subtracting the other member and vice versa:

como hacer ecuaciones de primer grado

We now group similar terms in each member, that is, we operate on one side with the terms that carry x and on the other side with the numbers:

como resolver ecuaciones de primer grado paso a paso

Now we can clear the x. To do this, she must be left alone. You have to remove the number you have in front of you, in this case 2, which, as it is multiplying, divides the other side of the equals:

ejercicios resueltos ecuaciones de primer grado

And now all that’s left to do is operate on the numbers side. In this case the fraction results in an exact number. If the fraction is not exact, it is simplified and left as a fraction:

ecuaciones de primer grado ejemplos

How to solve the first-degree equations with parentheses

I have to tell you, in order to solve the first-degree equations with parentheses, we only need to add one more step to the procedure we already know of solving first-degree equations:

  1. Delete parentheses
  2. Relocate terms: Passing terms with x to one member and numbers to the other member
  3. Simplify: Group similar terms together
  4. Clear x

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

We will see how to execute this new step and then we will 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 in front of it that is multiplying the terms within the parentheses.

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

In all these cases, the number must be multiplied by all the terms in the parentheses, taking into account the rule of the signs.

For example:

ecuaciones de primer grado con parentesis

ecuaciones con parentesis

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

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

There are more direct rules in both cases:

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

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

Solved exercises of first-degree equations with parentheses

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

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

ecuaciones de primer grado ejercicios resueltos

We eliminate the parentheses by multiplying the number in front of it by the members of the parentheses:

como resolver ecuaciones con parentesis

We have a much simpler first-degree equation, which has no parentheses, which we can continue to solve:

como resolver ecuaciones de primer grado con parentesis

como resolver ecuaciones con parentesis ejemplos

ecuaciones con parentesis ejercicios

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

How to solve first-degree equations with denominators

Now I will explain how to solve the first-degree denominator equations by following these steps:

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

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

In the next section, I will 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 denominators are eliminated.

We have the following equation:

ecuaciones de primer grado con denominadores

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

In this case, I will choose the 24, even if it is not the least common multiple, so that you will see that you can choose any denominator, as long as it is 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 with numbers only.

ecuaciones con denominadores

ecuaciones con denominador

In each member, we place everything in a single fraction, whose numerator is the sum or subtraction of all numerators.

At this point, it is when we can directly eliminate the denominator. Doing this is something similar to transposing terms. We can multiply both members by the common denominator, which is equivalent to eliminating them:

como quitar denominadores en una ecuación de primer grado

quitar denominadores en ecuaciones de primer grado

ecuaciones de primer grado con denominador

We have one first-degree equation left with parentheses.

We continue to eliminate the parentheses and end up solving the first degree equation:

ecuaciones de primer grado con denominadores

ecuacion con denominadores

como resolver ecuaciones con denominadores

ecuaciones de primer grado

The result is in the form of a fraction that we have had to simplify.

In the end, it is always resolved in the same way and that is to repeat the method over and over again. Once you’ve solved a few, it won’t be a problem.

To help you get 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

The first-degree equations with denominators are full of “pitfalls” that you have to take into account to solve them well. So let’s repeat another example to review once again the whole procedure, step by step and learn it well.

Let’s take the first example:

como resolver ecuaciones de primer grado paso a paso

We obtain common denominator and multiply the denominators with the corresponding numbers to transform their equivalent fractions into the original equation:

como hacer ecuaciones con denominadores

resolución de ecuaciones con denominadores

We group in a single fraction for each member and remove denominators:

ecuaciones lineales con denominador

como resolver ecuaciones de primer grado paso a paso

We eliminate parentheses and finish solving the equation:

elimina los denominadores y resuelve

ecuacion con denominador

ecuaciones con denominadores ejercicios

ecuaciones lineales con denominadores

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:

ecuaciones de primer grado con fracciones y paréntesis

First of all, we operate within the brackets to remove the parentheses. To do this, remember that the number multiplies each of the parenthesis terms:

ecuaciones con fracciones y parentesis

We now operate within the brackets grouping similar terms:

ecuaciones de primer grado con denominadores y parentesis

We eliminate the square brackets by multiplying the fraction. Remember that fractions multiply online. The terms enclosed in the square bracket have as denominator 1.

ecuaciones con denominadores y parentesis

We have terms with different denominator. We eliminate denominators, obtaining previously common denominator and calculating their equivalent fractions:

ecuaciones con parentesis y fracciones

Now we can remove the denominators. We’ve got this equation left:

ecuaciones con parentesis y denominadores

We continue to eliminate the parentheses:

ecuaciones de primer grado con parentesis y denominadores

If we have a minus sign preceding a fraction whose numerator has more than one term, remember that the minus sign affects all the numerator’s terms.

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

ecuaciones de primer grado con corchetes

We grouped terms together:

ecuaciones de primer grado con paréntesis y corchetes

And we clear the x:

ecuaciones de primer grado con denominadores

We have a fraction that cannot be simplified.

Here are the steps to solve first-degree equations with denominators and parentheses. The difficulty will depend on each equation.

How to solve first-degree equations with fractions

Next I will explain how the 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, where numerators have two or more terms.

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

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

Here are the steps to solve these types of equations:

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

Once we no longer have those brackets, the denominators can be removed normally and we can continue to solve 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 mentioned above, we may have fractions that are multiplying in parentheses and that we must remove before removing denominators.

We will explain it step by step with an example. This is a first-degree equation with fractions. Numerators have only one term:

ecuaciones de primer grado con fracciones

In which we have a fraction that is multiplying to a parenthesis:

ecuaciones con fracciones

We must begin by removing this parenthesis, because if we do not do so, it is not possible to remove denominators.

Remember that when two fractions are multiplying, they are multiplied in line, i. e. the numerator by the numerator and the denominator by the denominator.

Following the example, we multiply the parentheses and then operate in the denominators:

ecuaciones con fracciones paso a paso

como resolver ecuaciones de primer grado con fracciones

We’ve 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 lowest common multiple of 2,4,5 and 6:

m.c.m (2,3,5,6) = 30

como resolver ecuaciones con fracciones

ecuaciones de fracciones

ecuaciones con fracciones de primer grado

como hacer ecuaciones con fracciones

And now that we don’t have denominators, we operate on terms, relocate terms and clear the x as explained in the lesson on how to solve a first-degree equation:

ecuaciones de primer grado fracciones

como hacer ecuaciones de primer grado con fracciones

ecuación con fracciones

ecuaciones con fracciones y x

In this case the result cannot be simplified, but it must always be simplified.

For this is the procedure for solving first-degree equations with fractions.

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