Solve ##\dfrac{3x-6}{5-x}+\dfrac{11-2x}{10-4x}=3\dfrac{1}{2}##

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The discussion revolves around solving the equation 3(x-2)/(5-x) + (1/2)(11-2x)/(5-2x) = 3.5. The original poster found solutions x = 2 and x = 15/4 but noted that direct multiplication is tedious and suspected a simplification might exist. They highlighted the similarity of the denominators 5-x and 5-2x, indicating a potential for simplification that remains elusive. Participants emphasize a preference for methodical approaches over clever tricks, advocating for a solid understanding of basic principles in algebra. The conversation reflects a common challenge in algebraic manipulation and the search for efficient solving methods.
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Homework Statement
Solve ##\dfrac{3x-6}{5-x}+\dfrac{11-2x}{10-4x}=3\dfrac{1}{2}##
Relevant Equations
Algebraic manipulation
I've multiplied everything out on paper and got ##x=2, \dfrac{15}{4}##, which is correct. However multiplying directly is tedious and from observing this problem I suspect there is a simplification or trick that I missed.

##3\cdot\dfrac{x-2}{5-x}+\dfrac{1}{2}\cdot\dfrac{11-2x}{5-2x}=\dfrac{7}{2}##

Multiply both sides by ##2##:

##\dfrac{6(x-2)}{5-x}+\dfrac{11-2x}{5-2x}=7##

And then I'm stuck. But the denominators ##5-x, 5-2x## are tantalizingly close to each other, but I just can't figure out how to simplify/substitute/manipulate it to process this problem. Besides just multiplying it out of course.
 
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I don't see a way to simplify things. The best you could do is:
$$6(x-2)(5-2x) + (11-2x)(5-x) = 7(5-x)(5-2x)$$$$6(x-2)(5-2x) + (5-x)(11-2x - 35 + 14x) = 0$$$$6(x-2)(5-2x) + (5-x)(12x-24) = 0$$$$6(x-2)(5-2x) +6(x-2)(10 - 2x) = 0$$$$6(x-2)(15 - 4x) = 0$$
 
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<br /> -3-\frac{9}{x-5}+\frac{1}{2}-\frac{3}{2x-5}=3+\frac{1}{2}
\frac{3}{x-5}+\frac{1}{2x-5}=-2
4x^2-23x+30=0
(x-2)(4x-15)=0
 
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RChristenk said:
Homework Statement: Solve ##\dfrac{3x-6}{5-x}+\dfrac{11-2x}{10-4x}=3\dfrac{1}{2}##
Relevant Equations: Algebraic manipulation

I've multiplied everything out on paper and got ##x=2, \dfrac{15}{4}##, which is correct. However multiplying directly is tedious and from observing this problem I suspect there is a simplification or trick that I missed.

##3\cdot\dfrac{x-2}{5-x}+\dfrac{1}{2}\cdot\dfrac{11-2x}{5-2x}=\dfrac{7}{2}##

Multiply both sides by ##2##:

##\dfrac{6(x-2)}{5-x}+\dfrac{11-2x}{5-2x}=7##

And then I'm stuck. But the denominators ##5-x, 5-2x## are tantalizingly close to each other, but I just can't figure out how to simplify/substitute/manipulate it to process this problem. Besides just multiplying it out of course.
Here is my two cents.
I much prefer methodical approaches that use basic principles to reliably solve the vast majority of problems. I don't really care if they require a little more work. If you look for cute and clever tricks, you might learn a million of them and still not have a basic understanding.
 
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