Can someone help me solve for x and y

Question

Given $$ \left\{ \begin{array}{ll} x^4 y^3 &= e^{52} \\ \dfrac{x^3}{y^4} &= e^{14} \\ \end{array} \right. $$

Answer 1

HINT:

If a solution by algebra alone is preferred then let

$$ x^4 y^3 =p ,\dfrac{x^3}{y^4} = q. \tag 1 $$

By cubing the first, raising the second to fourth power and dividing we eliminate $x$

$$ y= \left(\frac{p^3}{q^4}\right)^{\frac{1}{25}} \tag2$$

Plug into either of 1) to find $x$.

Answer 2

Welcome to MSE. As a hint, you can take $\ln $ for both of equations. $$\left\{ \begin{array}{ll} x^4 y^3 &= e^{52} \to \ln(x^4 y^3)=\ln(e^{52})\\ \dfrac{x^3}{y^4} &= e^{14} \to \ln(\dfrac{x^3}{y^4})=\ln(e^{14})\\ \end{array} \right. $$ so you will have (for $a=\ln x,b=\ln(y)$)$$\left\{ \begin{array}{ll} 4a+3b=52 \\\ 3a-4b=14 \\ \end{array} \right. $$ canyou take over?

Answer 3

Remember the following three properties of logarithms:

1. $log(a\times{}b) = log(a) + log(b)$
2. $log(\frac{a}{b}) = log(a) - log(b)$
3. $log(a^b) = b\times{}log(a)$

Consider the first equation:

$$x^4y^3 = e^{52}$$

We can take the logarithm on both sides

$$log(x^4y^3) = log(e^{52})$$

Applying the first property

$$log(x^4) + log(y^3) = log(e^{52})$$

Applying the third property

$$4log(x) + 3log(y) = 52log(e)$$

Since we are using natural logarithms, $log(e) = 1$

$$4log(x) + 3log(y) = 52$$

Similarly, for the second equation:

$$\frac{x^3}{y^4} = e^{14}$$ $$log(\frac{x^3}{y^4}) = log(e^{14})$$ $$log(x^3) - log(y^4) = log(e^{14})$$ $$3log(x) - 4log(y) = 14log(e)$$ $$3log(x) - 4log(y) = 14$$

Now we have a pair of linear equations:

$$4log(x) + 3log(y) = 52$$ $$3log(x) - 4log(y) = 14$$

Multiply the first equation by 4, and the second equation by 3

$$16log(x) + 12log(y) = 208$$ $$9log(x) - 12log(y) = 42$$

Add both the equations

$$25log(x) = 250$$ $$log(x) = 10$$

By the definition of the logarithm:

$$x = e^{10}$$

Substitute $log(x) = 10$ into the first equation

$$40 + 3log(y) = 52$$ $$log(y) = 4$$ $$y = e^4$$

The values of x and y are $e^{10}$ and $e^4$ respectively