## anonymous one year ago 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.

1. anonymous

@whpalmer4, anyone?

2. anonymous

@sammixboo

3. anonymous

@prowrestler

4. anonymous

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

6. Michele_Laino

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

7. anonymous

is there a way of finding the intersection without graphing

8. 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: $\Large \left( {2,9} \right)$

9. Michele_Laino

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

10. Michele_Laino

computed*

11. anonymous

where did you get 2 from

12. Michele_Laino

I did some trial

13. anonymous

oh ok and thank you

14. 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"

15. anonymous

hold the question says use system of equations

16. Michele_Laino

yes! in fact I broke your equation in two functions

17. Michele_Laino

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

18. anonymous

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

19. Michele_Laino

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

20. anonymous

ok

21. anonymous

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

22. Michele_Laino

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

23. Michele_Laino

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

24. Michele_Laino

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

25. Michele_Laino

so we get the second intersection point

26. anonymous

ok thank you, no need to go further.

27. Michele_Laino

:)