- #1

RChristenk

- 49

- 4

- 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.

##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.