- anonymous

A biologist is comparing the growth of a population of flies per week to the number of flies a lizard will consume per week. She has devised an equation to solve for which day (x) the lizard would be able to eat the entire population. The equation is 3^x = 5x − 1.
Explain to the biologist how she can solve this on a graph using a system of equations.

- katieb

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- anonymous

@whpalmer4, anyone?

- anonymous

- anonymous

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## More answers

- anonymous

@Michele_Laino please please help me or ask someone for me

- Michele_Laino

A possible way is:
I call with g(x) the function 3^x, and with g(x) the function 5x-1, namely I write this:
\[\Large \begin{gathered}
f\left( x \right) = {3^x} \hfill \\
g\left( x \right) = 5x - 1 \hfill \\
\end{gathered} \]

- Michele_Laino

then I draw the graph of both functions f(x) and g(x) and I search for intersection point of those graphs

- anonymous

is there a way of finding the intersection without graphing

- Michele_Laino

one point is given setting x=2
we have:
\[\Large \begin{gathered}
f\left( 2 \right) = {3^2} = 9 \hfill \\
g\left( 2 \right) = 5 \cdot 2 - 1 = 9 \hfill \\
\end{gathered} \]
so the corresponding intersection point is:
\[\Large \left( {2,9} \right)\]

- Michele_Laino

another point can be compute, if we expand the function f(x) around x=0, using Taylor expansion

- Michele_Laino

computed*

- anonymous

where did you get 2 from

- Michele_Laino

I did some trial

- anonymous

oh ok and thank you

- Michele_Laino

please wait, try to write the Taylor expansion, around, x=0 of f(x), or try to use a software online like "desmos"

- anonymous

hold the question says use system of equations

- Michele_Laino

yes! in fact I broke your equation in two functions

- Michele_Laino

here is the system:
\[\Large \left\{ \begin{gathered}
f\left( x \right) = {3^x} \hfill \\
g\left( x \right) = 5x - 1 \hfill \\
\end{gathered} \right.\]

- anonymous

but there is no way of eliminating anything so is that why we graph

- Michele_Laino

for example, I write the Taylor expansion of f(x):

- anonymous

ok

- anonymous

it is just that i did not learn about the taylor expansion

- Michele_Laino

please here is the expansion up to the first order term:

- Michele_Laino

\[\Large {3^x} \simeq {\left. {{3^x}} \right|_{x = 0}} + {\left. {{3^x}\log 3} \right|_{x = 0}}x = 1 + x\log 3\]

- Michele_Laino

now, if we want to solve your equation, namely:
3^x=5x-1, we can solve this equation:
\[\Large 1 + x\log 3 = 5x - 1\]

- Michele_Laino

so we get the second intersection point

- anonymous

ok thank you, no need to go further.

- Michele_Laino

:)

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