Equation with variable in the denominator

In this article we will learn how to solve equations with an unknown in the denominator. We will start with a simple exercise and then move on.

$\frac{3}{X}=6$

So far we have not performed exercises with a variable in the denominator. In this case, the denominator of the left-hand member of the equation is $X$. How does this affect our exercise? This means that the exercise has a solution set (sometimes called a solution set), i.e. there are numbers that if we put them in place of the $X$ we will get a meaningless expression.

Whenever we go to solve an exercise with an unknown in the denominator, we will first find out what the solution set is, i.e., what number would set the denominator to zero and cause the expression to be meaningless.

In our case, the denominator is $X$, therefore, the solution set is

$X ≠0$

• well, the solution set is any $X$ as long as
  $X≠0$
  
  because if we put $X ≠0$ we will get an invalid expression (denominator $0$).

Now we will continue with the exercise. We will want to get rid of the denominator, so we will multiply both sides of the equation by $X$.

$3=6\times X$

and we will get

$3=6X$

$X=\frac{1}{2}$

Now, we must check that the solution belongs to the set. In our case

$X=\frac{1}{2}$

it does indeed belong to the solution set which is

$X ≠0$

therefore, $X=\frac{1}{2}$

is the solution.

At this stage it is highly recommended to check it in the following way. We will put $X=\frac{1}{2}$

in the original expression and verify it:

That is, the answer is correct.

$\frac{2}{X-1}=3$

First, we will write down the solution set. We must verify that the denominator differs from 0, therefore, we will write down that the solution set is:

$X-1 ≠0$

$X ≠1$

We advise to write it down clearly and prominently. We will return to this at the end of the exercise.

In the meantime we will write down:

Solution set: $X ≠1$

Now we will return to the solution set of the exercise. We will want to get rid of the denominator, therefore, we will multiply the two sides of the equation by $X-1$

We will obtain:
$(\frac{2}{X-1}=3)\times (X-1)$

$2=3\times (X-1)$

$2=3X-3$

$5=3X$

$X=\frac{5}{3}$

Now we will check if the solution is in the set. Recall that the solution set is

$X ≠1$

i.e., the solution we have obtained is in it, therefore, the result is $X=\frac{5}{3}$

At this stage it is strongly recommended to place the result in the original expression to see if we got it right. Try it!

$\frac{2}{X-3}-3=2$

First, we will find out what the solution set is.

$X-3 ≠0$

$X ≠3$

Now we go back to the exercise. We pass the 3 to the other side and continue solving. We will get

$\frac{2}{X-3}=5$

Now we want to get rid of the denominator. We'll multiply both sides of the equation by the denominator and we'll get:

$(\frac{2}{X-3}=5)\times (X-3)$

$2=5\times (x-3)$

$2=5X-15$

$5X=17$

$X=\frac{17}{5}$

This solution effectively lies within the solution set which is. $X ≠3$

It is recommended to verify that the result is correct by placing it in the original expression.

$\frac{2}{X-3}=\frac{1}{X}$

We will notice that in this exercise there is more than one denominator. First, we will want to find out what the solution set is. We will find out separately for each denominator.

That is, we must first check that:

$X-3 ≠0$
$X ≠3$
In addition, we have to check the denominator located on the right side of the equation, that is:

$X≠0$

Before continuing with the resolution of the exercise we will write it down in a clear and orderly manner.
The solution set is
$X ≠3$
$X ≠0$

Now we will want to get rid of both denominators. To do so, we will multiply both members of the equation by the common denominator. The common denominator in this case is $X\times (X-3)$

$(\frac{2}{X-3}=\frac{1}{X})\times X(X-3)$

$\frac{2}{X-3}\times X(X-3)=\frac{1}{X}\times X(X-3)$

$2X=X-3$
$X=-3$

The result $X=-3$ is compatible with the solution set.

To corroborate our result, we will place it in the original equation and examine if a true equality is obtained:

$\frac{1}{(-3)}=\frac{2}{(-3-3)}$

$\frac{1}{(-3)}=\frac{1}{(-3)}$

Then, our result is correct.

$\frac{3}{4X}+2=\frac{1}{8X}$

First we will find out which is the solution set. We must verify that both denominators differ from zero. In this case the number 0 will set both denominators to zero, so we will write down:

The solution set is $X\ne0$
Also in this exercise we have two denominators with unknown and we will want to get rid of both. To do this we will multiply them by the common denominator. We will notice that if we multiply $4X$ by $2$ we will get $8X$.
Therefore, the common denominator is $8X$

$(\frac{3}{4X}+2=\frac{1}{8X})\times 8X$

$\frac{3}{4X}\times 8X+2\times 8X=\frac{1}{8X}\times 8X$

$6+16X=1$
$16X=-5$

$X=-\frac{5}{16}$

The result we obtained fits the solution set.
It is highly recommended to place it in the original exercise to corroborate that a true equality is obtained.

$\frac{3}{4\times (-\frac{5}{16})}+2=\frac{1}{8\times (-\frac{5}{16})}$

$\frac{3}{(-\frac{5}{4})}+2=\frac{1}{(-\frac{5}{2})}$

$-\frac{12}{5}+2=-\frac{2}{5}$

$12-10=2$

$2=2$

We have obtained a true equality, therefore, our solution is correct.

$\frac{3}{3X-2}+1=\frac{1}{6X-4}$
We will notice that the denominator on the right side can be divided into other terms by taking out the common term.

We will do it and we will obtain:

$\frac{3}{3X-2}+1=\frac{1}{2(3X-2)}$

Now we will continue with the exercise. First we will find the solution set. We will write down

$3X-2 ≠0$

$3X ≠2$

$X ≠\frac{2}{3}$

Let's go back to the exercise. To get rid of the denominator we have to multiply both members of the equation by the common denominator. As we can see, after taking out the common term, it gives us that the common denominator is $2\times (3X-2)$

Therefore, we will multiply both members of the equation by it and we will get

$(\frac{3}{3X-2}+1=\frac{1}{2(3X-2)})\times 2(3X-2)$

$\frac{3}{3X-2}\times 2\times(3X-2)+1 \times 2 \times(3X-2)=\frac{1}{2\times(3X-2)}\times 2\times(3X-2)$

$6+6X-4 =1$

$6X=-1$

$X=-\frac{1}{6}$

We will see that the result obtained is indeed in the solution set. $X ≠\frac{2}{3}$

At this stage it is strongly recommended to place the result obtained in the original equation to check if we have done it right. Try it!