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# Equation with the variable in the denominator

Sal solves the equation 7 - 10/x = 2 + 15/x. 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?
• 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
18 = 9x
Divide both sides by 9.
2 = x
• just multiply everything by that variable to cancel it out from the denominator and then solve for that variable
• If 19=43+X what is the value of X
• 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.
• In the video, Sal minused the 2x, but couldn't you also minus the 7x to? I
(1 vote)
• 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.

``6/x + 3 = 12/2x + 1``

We can try to solve it as usual:

``Let's multiply both sides on x6 + 3x = 12/2 + xThen isolate x (subtract it from the both sides)6 + 2x = 12/2This is equals6 + 2x = 6Subtract 6 from the both sides2x = 0Divide by 2 both sidesx = 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?
• 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.
• 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
• 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?
• Remember that "x" is really "x/1," just like 2 is 2/1.
Let's take a similar problem without variables: 2 &bull; 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 &bull; 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.