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

[RChristenk]

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.

 

[PeroK]

I don't see a way to simplify things. The best you could do is:
$$6(x-2)(5-2x) + (11-2x)(5-x) = 7(5-x)(5-2x)$$$$6(x-2)(5-2x) + (5-x)(11-2x - 35 + 14x) = 0$$$$6(x-2)(5-2x) + (5-x)(12x-24) = 0$$$$6(x-2)(5-2x) +6(x-2)(10 - 2x) = 0$$$$6(x-2)(15 - 4x) = 0$$

 

[anuttarasammyak]

$$-3-\frac{9}{x-5}+\frac{1}{2}-\frac{3}{2x-5}=3+\frac{1}{2}$$
$$\frac{3}{x-5}+\frac{1}{2x-5}=-2$$
$$4x^2-23x+30=0$$
$$(x-2)(4x-15)=0$$

 

[FactChecker]

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.