Mathematics: Which determinants can be used to solve for x and y in the system of linear equations below? 4x+16y=-68 12x+3y=21

Which determinants can be used to solve for x

Ответ:

2025hollowaykyva

14.03.2022

Are there choices? If there are, you should list them because there are some equivalents.
$\left[\begin{array}{cc}4&16\\12&3\end{array}\right] \\ \left[\begin{array}{cc}-68&16\\21&3\end{array}\right] = x \\ \left[\begin{array}{cc}4&-68\\12&21\end{array}\right] = y \\$

I'm assuming that you can solve these things. The top determinant comes to -180. The x value comes to - 540. When divided by the top one x = 3

The bottom determinant (y) comes to 900 and when divided by - 180 makes y = - 5. The checks out with actually solving the system of equations some other way.

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Ответ:

sihamabdalla591

25.03.2023

Mathematics

First a review of log properties:
log (ab) = log(a) + log(b)
log (a/b) = log(a) - log(b)
log (a^n) = n*log(a)

In this way by taking the log of entire function we can break it up into more simple terms before trying to take the derivative.

But we must take log of BOTH sides.
$y = \frac{x^7 (x-6)^8}{(x^2+5)^9} \\ ln (y) = ln (\frac{x^7 (x-6)^8}{(x^2+5)^9})$
Expand Right side
$ln(y) = ln (x^7) + ln (x-6)^8 - ln (x^2+5)^9 \\ ln (y) = 7 ln (x) +8 ln (x-6) - 9 ln(x^2 +5)$
Now lets review derivative of natural log:
$\frac{d}{dx} ln( u(x)) = \frac{1}{u(x)} \frac{du}{dx}$
Take derivative of both sides:
$\frac{1}{y} \frac{dy}{dx} = 7(\frac{1}{x})(1) + 8(\frac{1}{x-6})(1) - 9(\frac{1}{x^2+5})(2x) \\ \frac{1}{y} \frac{dy}{dx} = \frac{7}{x} + \frac{8}{x-6} - \frac{18x}{x^2+5}$
Solve for dy/dx by multiplying by "y". (Note y is original function)
$\frac{dy}{dx} = y(\frac{7}{x} + \frac{8}{x-6} - \frac{18x}{x^2+5}) \\ \frac{dy}{dx} = (\frac{x^7 (x-6)^8}{(x^2+5)^9})(\frac{7}{x} + \frac{8}{x-6} - \frac{18x}{x^2+5})$

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Which determinants can be used to solve for x and y in the system of linear equations below? 4x+16y=-68 12x+3y=21

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