Find an equation of the graph that passes through the point (7, 3) and has the slope $y ^ { t } = \frac { 5 y } { 2 x }$
A)$y = 3 ( 7 x ) ^ { \frac { 2 } { 5 } }$
B)$y = x e ^ { \frac { 5 } { 2 } + \frac { 7 } { 3 } }$
C)$y = 3 \left( \frac { x } { 7 } \right) ^ { \frac { 5 } { 2 } }$
D)$y = 7 \left( \frac { 5 x } { 2 } \right) ^ { 3 }$
E) $y = \frac { 2 } { 5 x } - \ln ( 7 x ) + 3$

Question

Quizplus · Accepted Answer

Firstly, it is necessary to re-arrange the given slope equation in order to make it easier to work with:
$$y^{t} = \frac{5y}{2x}$$
$$2xy^{t} = 5y$$
$$y^{1-t} = \frac{2x}{5}$$

Since we know the point (7,3) lies on the graph, we can use this to solve for the constant in our final equation. Plugging in x = 7 and y = 3, we have:

$$3^{1-t} = \frac{2 \cdot 7}{5}$$
$$3^{1-t} = \frac{14}{5}$$
$$1-t = \log_{3} \frac{14}{5}$$
$$t = 1 - \log_{3} \frac{14}{5}$$

Therefore, the final equation of the graph is:
$$y = 3 \left(\frac{x}{7}\right)^{1-t}$$
$$y = 3 \left(\frac{x}{7}\right)^{\log_{3} \frac{14}{5}}$$
$$\boxed{y = 3 \left(\frac{x}{7}\right)^{\frac{5}{\log_{3} 14 - \log_{3} 5}}}$$

This matches choice C, which is the correct answer.

Find an Equation of the Graph That Passes Through the Point