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solve this equation

2/x+4=5/3x+5

 Nov 10, 2017
 #1
avatar+537 
0

Here is the answer, but all of the work may take a little to type in.

 Nov 10, 2017
 #2
avatar
0

Solve for x:
4 + 2/x = 5 + 5/(3 x)

5 + 5/(3 x) = 5 + 5/(3 x):
4 + 2/x = 5 + 5/(3 x)

Bring 4 + 2/x together using the common denominator x. Bring 5 + 5/(3 x) together using the common denominator 3 x:
(2 (2 x + 1))/x = (5 (3 x + 1))/(3 x)

Cross multiply:
6 x (2 x + 1) = 5 x (3 x + 1)

Expand out terms of the left hand side:
12 x^2 + 6 x = 5 x (3 x + 1)

Expand out terms of the right hand side:
12 x^2 + 6 x = 15 x^2 + 5 x

Subtract 15 x^2 + 5 x from both sides:
x - 3 x^2 = 0

Factor x and constant terms from the left hand side:
-(x (3 x - 1)) = 0

Multiply both sides by -1:
x (3 x - 1) = 0

Split into two equations:
x = 0 or 3 x - 1 = 0

Add 1 to both sides:
x = 0 or 3 x = 1

Divide both sides by 3:
x = 0 or x = 1/3

4 + 2/x ⇒ 2/0 + 4 = ∞^~
5 + 5/(3 x) ⇒ 5/(3 0) + 5 = ∞^~:
So this solution is incorrect

4 + 2/x ⇒ 4 + 2/(1/3) = 10
5 + 5/(3 x) ⇒ 5 + 5/(3/3) = 10:
So this solution is correct

The solution is: x = 1/3

 Nov 10, 2017
 #3
avatar+537 
-1

Are you sure?

ProMagma  Nov 10, 2017
edited by ProMagma  Nov 10, 2017
 #4
avatar+537 
0

I thought I got it, but he has a lot more evidence...

ProMagma  Nov 10, 2017
 #5
avatar+2340 
+2

Upon further review, there appears to be ambiguity because the interpretations of 2/x+4=5/3x+5 are different.

 

ProMagma solved the equation \(\frac{2}{x}+4=\frac{5}{3}x+5\)

 

Guest solved the equation \(\frac{2}{x}+4=\frac{5}{3x}+5\)

 

This discrepancy occurs at the right side of the equation. That division symbol causes interpretations to differ; there really should be parentheses to clarify. This would not be the first time this forum has seen this issue. Strictly speaking, I believe that ProMagma's interpretation to be correct, but ManuelBautista2019 could have meant the other interpretation.

 Nov 10, 2017
 #7
avatar+537 
0

Thank You :)

ProMagma  Nov 12, 2017
 #6
avatar+1405 
0

Thank you Promagna, Guest, XSquaredFactor

 Nov 10, 2017

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