Find the slope of the line passing through the points (-3,7) and (2,-6)

Mathematics

1 answer:

elena55 [62]3 years ago

7 0

Answer:

$-\frac{13}{5}$

Step-by-step explanation:

by using formula $\frac{y_2-y_1}{x_2-x_1}$ , we get

$\frac{-6-7}{2-(-3)}=-\frac{13}{5}$

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Given that 3^x = 4^y = 12^z, show that z = (xy)/(x+y).

irinina [24]

$3^{x} = 4^{y} = 12^{z}$
$3^{x} = 4^{y} = (4 \cdot 3)^{z}$
$3^{x} = 4^{y} = 4^{z} \cdot 3^{z}$

$\text{Let a } = 3^{x} = 4^{y} = 4^{z} \cdot 3^{z}$
$log_3a = x$
$log_4a = y$
$log_{(4 \cdot 3)}a = z$

Using change of base:
$x = \frac{lna}{ln3}$
$y = \frac{lna}{ln4}$
$z = \frac{lna}{ln(4 \cdot 3)}$

$ln3 = \frac{lna}{x}$
$ln4 = \frac{lna}{y}$
$ln(4 \cdot 3) = \frac{lna}{z}$

Now, ln(4 · 3) = ln(4) + ln(3)

$\frac{lna}{z} = \frac{lna}{x} + \frac{lna}{y}$
$\frac{1}{z} = \frac{1}{x} + \frac{1}{y}$
$\frac{1}{z} = \frac{x + y}{xy}$

$\therefore z = \frac{xy}{x + y}$