How do you solve for u in $\frac{4}{u + 6} = \frac{6}{u + 6} + 2$ $4/(u+6)=6/(u+6)+2$ ?

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How do you solve for u in $\frac{4}{u + 6} = \frac{6}{u + 6} + 2$ $4/(u+6)=6/(u+6)+2$ ?

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Answer 1 · 2018-01-11T08:20:30

$u = - 7$ $u=-7$

Explanation:

First, we need to put the requirements, that is $u \neq - 6$ $u!=-6$ because then the denominator will be $0$ $0$ , and make the equation undefined.

Then,

$\frac{4}{u + 6} = \frac{6}{u + 6} + 2$ $4/(u+6)=6/(u+6)+2$

$- \frac{2}{u + 6} = 2$ $-2/(u+6)=2$

$u + 6 = - 1$ $u+6=-1$

$u = - 7$ $u=-7$

Answer 2 · 2018-02-08T21:28:19

A different approach: $u = - 7$ $u=-7$

Explanation:

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

$\frac{4}{u + 6} = \frac{6}{u + 6} + 2 d \to d \frac{4}{u + 6} = \frac{6}{u + 6} + [2 \times 1]$ $color(green)(4/(u+6)=6/(u+6)+2 color(white)("d")->color(white)("d")4/(u+6)=6/(u+6)+[2color(red)(xx1)])$

$ddddddddddddddddd \to d \frac{4}{u + 6} = \frac{6}{u + 6} + [2 \times \frac{u + 6}{u + 6}]$ $color(green)( color(white)("ddddddddddddddddd")->color(white)("d")4/(u+6)=6/(u+6)+[2color(red)(xx(u+6)/(u+6))])$

$dDDDDddddddddddd \to d \frac{4}{u + 6} = \frac{6}{u + 6} + d \frac{2 (u + 6)}{u + 6}$ $color(green)(color(white)("dDDDDddddddddddd")->color(white)("d")4/(u+6)=6/(u+6)+color(white)("d")(2(u+6))/(u+6)$

Now all the denominators (bottom numbers) are the same we can forget about them.

Or, as a purist would say: multiply all of both sides by $(u + 6)$ $(u+6)$ . This cancels out the denominators which is THE SAME THING!

$4 = 6 + 2 (u + 6)$ $color(green)(4=6+2(u+6))$

$4 = 6 + 2 u + 12$ $color(green)(4=6+2u+12)$

$4 = 18 + 2 u$ $color(green)(4=18+2u)$

Subtract 18 from both sides

$2 u = - 14$ $color(green)(2u=-14)$

Divide both sides by 2

$u = - 7$ $color(green)(u=-7)$

$Foot note: as u can only take on one value for this to work$ $color(blue)("Foot note: as "u" can only take on one value for this to work ")$ $and this is not -6 then we do not need to state that x \neq - 6$ $color(blue)("and this is not -6 then we do not need to state that "x!=-6)$

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How do you solve for u in $\frac{4}{u + 6} = \frac{6}{u + 6} + 2$ $4/(u+6)=6/(u+6)+2$ ?

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