How do you simplify the expression (y-12/(y-4))/(y-18/(y-3))?

Jul 2, 2017

I chose to multiply by a form of 1 that eliminates the complex fractions but there may be a better way, because I, later, discover a common factor that becomes 1.

Explanation:

Simplify: $\frac{y - \frac{12}{y - 4}}{y - \frac{18}{y - 3}}$

Multiply the expression by 1 in the form of $\frac{\left(y - 4\right) \left(y - 3\right)}{\left(y - 4\right) \left(y - 3\right)}$:

$\frac{y - \frac{12}{y - 4}}{y - \frac{18}{y - 3}} \frac{\left(y - 4\right) \left(y - 3\right)}{\left(y - 4\right) \left(y - 3\right)}$

Multiply the two fractions:

$\frac{\left(y \left(y - 4\right) \left(y - 3\right)\right) - 12 \left(y - 3\right)}{\left(y \left(y - 4\right) \left(y - 3\right)\right) - 18 \left(y - 4\right)}$

Substitute $\left({y}^{2} - 7 y + 12\right)$ for (y-4)(y-3)):

$\frac{y \left({y}^{2} - 7 y + 12\right) - 12 \left(y - 3\right)}{y \left({y}^{2} - 7 y + 12\right) - 18 \left(y - 4\right)}$

Distribute y:

$\frac{{y}^{3} - 7 {y}^{2} + 12 y - 12 \left(y - 3\right)}{{y}^{3} - 7 {y}^{2} + 12 y - 18 \left(y - 4\right)}$

Distribute 12 and 18:

$\frac{{y}^{3} - 7 {y}^{2} + 12 y - 12 y + 36}{{y}^{3} - 7 {y}^{2} + 12 y - 18 y + 72}$

Combine like terms:

$\frac{{y}^{3} - 7 {y}^{2} + 36}{{y}^{3} - 7 {y}^{2} - 6 y + 72} \text{ [1]}$

I asked WolframAlpha to factor the numerator and obtained the following answer:

$\left(y + 2\right) \left(y - 3\right) \left(y - 6\right)$

Then I asked WolframAlpha to factor the denominator and obtained the following answer:

$\left(y + 3\right) \left(y - 4\right) \left(y - 6\right)$

Please observe that $\left(y - 6\right)$ is common to both numerator and denominator, therefore, expression [1] becomes:

$\frac{\left(y + 2\right) \left(y - 3\right)}{\left(y + 3\right) \left(y - 4\right)}$

Jul 2, 2017

$\frac{y - \frac{12}{y - 4}}{y - \frac{18}{y - 3}} = \frac{\left(y + 2\right) \left(y - 3\right)}{\left(y + 3\right) \left(y - 4\right)}$

Explanation:

We want to simplify:

$\frac{y - \frac{12}{y - 4}}{y - \frac{18}{y - 3}}$

Firstly consider the numerator, which we can simplify by putting over a common denominator, thus:

$y - \frac{12}{y - 4} = \frac{y \left(y - 4\right) - 12}{y - 4}$
$\text{ } = \frac{{y}^{2} - 4 y - 12}{y - 4}$
$\text{ } = \frac{\left(y + 2\right) \left(y - 6\right)}{y - 4}$

Secondly consider the denominator, which we can also simplify by putting over a common denominator, thus:

$y - \frac{18}{y - 3} = \frac{y \left(y - 3\right) - 18}{y - 3}$
$\text{ } = \frac{{y}^{2} - 3 y - 18}{y - 3}$
$\text{ } = \frac{\left(y + 3\right) \left(y - 6\right)}{y - 3}$

So now we can rewrite the expression as:

$\frac{y - \frac{12}{y - 4}}{y - \frac{18}{y - 3}} = \frac{\frac{\left(y + 2\right) \left(y - 6\right)}{y - 4}}{\frac{\left(y + 3\right) \left(y - 6\right)}{y - 3}}$

$\text{ } = \left\{\frac{\left(y + 2\right) \left(y - 6\right)}{y - 4}\right\} \cdot \left\{\frac{y - 3}{\left(y + 3\right) \left(y - 6\right)}\right\}$

Then cancelling the common factor of $\left(y - 6\right)$ we get:

$\frac{y - \frac{12}{y - 4}}{y - \frac{18}{y - 3}} = \left\{\frac{\left(y + 2\right)}{y - 4}\right\} \cdot \left\{\frac{y - 3}{\left(y + 3\right)}\right\}$

$\text{ } = \frac{\left(y + 2\right) \left(y - 3\right)}{\left(y + 3\right) \left(y - 4\right)}$

Jul 2, 2017

color(magenta)(((y+2)(y-3))/((y+3)(y-4))

Explanation:

$\frac{y - \frac{12}{y - 4}}{y - \frac{18}{y - 3}}$

$\therefore = \frac{\frac{y \left(y - 4\right) - 12}{y - 4}}{\frac{y \left(y - 3\right) - 18}{y - 3}}$

$\therefore = \frac{{y}^{2} - 4 y - 12}{y - 4} \times \frac{y - 3}{{y}^{2} - 3 y - 18}$

:.=((y+2)(cancel(y-6)))/((y-4)) xx ((y-3))/((y+3)(cancel(y-6))

:.=color(magenta)(((y+2)(y-3))/((y+3)(y-4))