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### Course: 8th grade (Eureka Math/EngageNY) > Unit 4

Lesson 1: Topic A: Writing and solving linear equations- Intro to equations with variables on both sides
- Equations with variables on both sides: 20-7x=6x-6
- Equations with variables on both sides
- Equation with variables on both sides: fractions
- Equations with variables on both sides: decimals & fractions
- Equation with the variable in the denominator
- Equations with parentheses
- Equations with parentheses
- Equations with parentheses: decimals & fractions
- Number of solutions to equations
- Worked example: number of solutions to equations
- Number of solutions to equations
- Creating an equation with no solutions
- Creating an equation with infinitely many solutions
- Number of solutions to equations challenge

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# Equation with the variable in the denominator

The variable x is in the denominator in 7 - 10/x = 2 + 15/x. We can multiply both sides of the equation by x to turn it into a more familiar, linear form without fractions. From there, we isolate the variable to solve. P.S. x cannot equal zero, because then the equation would be undefined. Created by Sal Khan.

## Want to join the conversation?

- What if you have a fraction in the equation that's like : 7/x-9 and the other side of the equation has -2/x, how would you work that out?(19 votes)
- You will cross multiply and then solve. To cross multiply, multiply the denominator on the left by the numerator on the right. Then do the same for the denominator on the right and the numerator on the left. These new expressions will be equal to each other. For your example, it will look like:

(x-9)(-2)=(x)(7)

Distribute/multiply to get:

-2x + 18 = 7x

Add 2x to both sides to gather your variables:

18 = 9x

Divide both sides by 9.

2 = x(25 votes)

- What about this equation?
`6/x + 3 = 12/2x + 1`

We can try to solve it as usual:`Let's multiply both sides on x`

6 + 3x = 12/2 + x

Then isolate x (subtract it from the both sides)

6 + 2x = 12/2

This is equals

6 + 2x = 6

Subtract 6 from the both sides

2x = 0

Divide by 2 both sides

x = 0

But we can't substitute zero instead of`x`

in the equation because it's impossible to have zero as denominator.

So should we care about exclusions for`x`

when it resides in denominator?(13 votes)- Yes, you should care about exclusions for X. While your work created x=0, it is not a valid solution because it doesn't make the equation be true (both sides equal). Division by 0 is undefined.(14 votes)

- if you keep pushing 1 on ur keyboard he says boat boat boat boat(16 votes)
- If 19=43+X what is the value of X(4 votes)
- The answer is negative 24, because 43+(-24)=19, or you can solve it by isolating the x, to do this, put the-equation X+43=19, subtract 43 from both sides so it will look like, (X+43)-43=(19)-43, so (x+43)-43 is X, (19)-43 is -24, so X=-24.

Hope this helps.(25 votes)

- I got the right answer too when I added 10/x both sides of the equation.

7-10/x=2+15/x

7+10/x-10/x=2+10/x+15/x

7=2+25/x

-2+7=-2+2+25/x

5=25/x

5=x(15 votes)- Nice! Now you gotta do that multiple times...(2 votes)

- In the video, Sal minused the 2x, but couldn't you also minus the 7x to? I(2 votes)
- Yes, you could have subtracted 7x rather than the 2x. If you prefer to work with positive numbers, then moving the 2x keeps the coefficient of x a positive value.(24 votes)

- What if the numerator on both sides is x and the denominator of both fractions does not have a common multiple? Could we still solve for x using this method or some other type?(6 votes)
- There's always a common multiple of the denominators. The product of the denominators is always a common multiple, though isn't always the least common multiple.

Have a blessed, wonderful day!(3 votes)

- Why does multiplying x by 10/x get you 10? Is it because if u multiply you get 10x over x which simplifies into 10?(4 votes)
- Remember that "x" is really "x/1," just like 2 is 2/1.

Let's take a similar problem without variables: 2 • 3/2.

When you multiply 2 (or 2/1) by 3/2, you multiply numerator by numerator, and denominator by denominator. You end up with 6/2. When you reduce (or simplify), you divide both the numerator and the denominator by their GCF (greatest common factor). 6/2 = 3, and 2/2 = 1.

So you're left with 3/1, or**3**.

Now look back at your original problem, x • 10/x. When you multiply (remember that x = x/1), you end up with 10x/x. Now we need to simplify. Obviously the only factor between the top and bottom is x, so we divide both the numerator and the denominator by x.

10x/x = 10, and x/x = 1,

so we're left with 10/1, or**10**.(5 votes)

- so all we really have to do is distribute the x and then go from there?(7 votes)
- What is 7Xsomething is7(3 votes)
- Do you mean 7 * x = 7, if so, then x = 1(5 votes)

## Video transcript

So I have the equation 7 minus
10/x is equal to 2 plus 15/x. And so this isn't
the type of equation that you might think that
you're used to solving. But I'll give you a
few moments to see if you can solve it on your own. Well, what we'll see is we
can do a quick multiplication of both sides to actually
simplify this to a form that we are more
used to looking at. So what's probably bothering
you, because it's bothering me, is these x's that we have in the
denominators right over here. We're like, well, how
do we deal with that? Well, whenever we see
an x in the denominator, the temptation is
to multiply it by x. But we can't just multiply
one of the terms by x. We have to multiply
the entire side by x. So we could multiply
this entire side by x. But we can't just multiply
the left-hand side by x. We'd also want to multiply
the right-hand side by x. And so what will that give us? Well, we distribute the x. We get x times 7 is 7x. And then x times
negative 10/x, well, that's just going
to be negative 10. So you get negative
10 right over there. So the left-hand side
simplifies to 7x minus 10. And then your right-hand
side, once again, distribute the x.
x times 2 is 2x. x times 15/x, well,
x times something over x is just going to be
the something. x times 15/x is just going to
be 15-- plus 15. So now we've simplified
this to a linear equation. We have the variable
on both sides. So we just have to do
some of the techniques that we already know. So the first thing
that I like to do is maybe get all my x's
on the left-hand side. So I want to get rid of
this 2x right over here. So I subtract 2x from
the right-hand side. Now, and I always remind
you, I can't do that just to the right-hand side. If I did it just to
the right-hand side, it wouldn't be an
equality anymore. You have to do that to the
left-hand side as well. And so we are left with-- let
me get that pink color again. On the left-hand side, 7x, 7 of
something minus 2 of something, well, you're going to have 5
of that something, minus 10. These two x's negate each other. And you're left with equals 15. Now we can get rid
of this negative 10 by adding 10 to both sides. You know, I like
that green color when I do stuff to both sides. So I can add 10 to both sides. And I'm left with
5x-- these negate each other-- is equal to 25. And this is the home stretch. You see where this is going. We can divide both sides by 5. And we are left with
x is equal to 5. Now let's verify that
this actually worked. So let's go back to
the original equation. We have 7 minus 10/5. This needs to be
equal to-- I'm just taking our 5 and
substituting it back here. This needs to be
equal to 2 plus 15/5. So this is 7 minus 10/5. This is just 2. It needs to be equal to 2
plus 15/5, which is just 3. So 2 plus 3, 7 minus
2 is 5, 2 plus 3 is 5, 5 is indeed equal to 5. And we are done.