How do you solve (200/r)-1= 200/(r-10)?

Apr 5, 2018

$r = 5 \pm \frac{\sqrt{- 7900}}{2}$

Explanation:

$\frac{200}{r} - 1 = \frac{200}{r - 10}$

Rearrange the equation

$\frac{200}{r} - \frac{200}{r - 10} = 1$

Take $200$ common

$200 \left[\frac{1}{r} - \frac{1}{r - 10}\right] = 1$

Bring $200$ to other side

$\frac{1}{r} - \frac{1}{r - 10} = \frac{1}{200}$

Make denominators equal

(1/r × (r - 10)/(r - 10)) - (1/(r - 10) xx r/r) = 1/200

$\frac{r - 10}{r \left(r - 10\right)} - \frac{r}{r \left(r - 10\right)} = \frac{1}{200}$

Now, denominators are same. Numerators can be added

$\frac{r - 10 - r}{r \left(r - 10\right)} = \frac{1}{200}$

$\frac{- 10}{r \left(r - 10\right)} = \frac{1}{200}$

$r \left(r - 10\right) = - 2000$

${r}^{2} - 10 r = - 2000$

r^2 - 10r + 2000 = 0 color(white)(.)……[1]

Use quadratic equation formula to find value(s) of $r$

$r = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

In equation $\left[1\right]$

• $a = 1$

• $b = - 10$

• $c = 2000$

r = (-(-10) +- sqrt((-10)^2 - (4 × 1 × 2000)))/(2 × 1)

$r = \frac{10 \pm \sqrt{100 - 8000}}{2}$

$r = \frac{10 \pm \sqrt{- 7900}}{2}$

$r = 5 \pm \frac{\sqrt{- 7900}}{2}$