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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.

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• 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
Add 2x to both sides to gather your variables:
18 = 9x
Divide both sides by 9.
2 = x
• In the video, Sal minused the 2x, but couldn't you also minus the 7x to? I
• 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.
• If 19=43+X what is the value of X
(1 vote)
• 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.
• 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.
• Wait but when you multiply the problem through by x like Sal did at , this is the same thing as (x/1) which is not equivalent to 1--so Sal is changing the equation. How does this work?
• The properties of equality let us change equations into equivalent equations as long as we do the same operation to both side. Sal is multiplying both sides of the equation by "x". So, his version of the equation is equivalent to the original.
Hope this helps.
• Around , Sal says this simplifies to a linear equation. Doesn't a linear equation require that there be two variables?
• A linear equation can have 1 variable. It then creates a horizontal or vertical line.
For example, in the video, Sal gets "x=5". This creates a vertical line. It is interpreted as x=5 for all values of Y. So, the line contains points like: (5, 0); (5,-3); (5,11); etc.
Hope this helps.

``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.
• What is 7Xsomething is7