Question

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.

Answer 1

@whpalmer4, anyone?

Answer 2

@sammixboo

Answer 3

@prowrestler

Answer 4

@Michele_Laino please please help me or ask someone for me

Answer 5

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

Answer 6

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

Answer 7

is there a way of finding the intersection without graphing

Answer 8

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:
\[\Large \left( {2,9} \right)\]

Answer 9

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

Answer 10

computed*

Answer 11

where did you get 2 from

Answer 12

I did some trial

Answer 13

oh ok and thank you

Answer 14

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

Answer 15

hold the question says use system of equations

Answer 16

yes! in fact I broke your equation in two functions

Answer 17

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

Answer 18

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

Answer 19

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

Answer 20

ok

Answer 21

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

Answer 22

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

Answer 23

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

Answer 24

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

Answer 25

so we get the second intersection point

Answer 26

ok thank you, no need to go further.

Answer 27

:)