Question

How do you solve `(5x-10)/(7x+6)=10/8`?

Answer 1

I got `x = -14/3`.

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I would first multiply by `8`, and then by `7x+6`.

`(5x - 10)/(7x + 6) = 10/8`

`(8*(5x - 10))/(7x + 6) = 10/cancel(8)*cancel(8)`

`(8(5x - 10))/cancel((7x + 6))*cancel((7x + 6)) = 10*(7x + 6)`

From here, just distribute the terms, move the same types of terms to each side, and solve for `x`.

`ul(8)(ul(color(red)(5))x - ul(color(darkblue)(10))) = ul(10)(ul(color(red)(7))x + ul(color(darkblue)(6)))`

`color(red)(40)x - color(darkblue)(80) = color(red)(70)x + color(darkblue)(60)`

`-140 = 30x`

`color(blue)(x) = -140/30`

`= color(blue)(-14/3)`

And we can prove that this is correct:

`(5(-14/3) - 10)/(7(-14/3) + 6) stackrel(?" ")(=) 10/8`

`(-70/3 - 30/3)/(-98/3 + 18/3) stackrel(?" ")(=) 10/8`

`(-100/3)/(-80/3) stackrel(?" ")(=) 10/8`

`color(red)(cancel(color(black)(-)))100/color(red)(cancel(color(black)(3)))*color(red)(cancel(color(black)(-)))color(red)(cancel(color(black)(3)))/80 stackrel(?" ")(=) 10/8`

`100/80 = color(green)(10/8 = 10/8)`