Mr. Torrey draws and labels a figure, as shown below. He asks a group of students to each write an expression that is equivalent to the perimeter of the figure in terms of the given side lengths. Clockwise from the top, the figure reads, 5 r, 2 s, 3 r, 2 s, 5 r, 2 s, 3 r, 2 s.

The students’ expressions are shown below.

Brian: $8\left(2r\ +\ s\right)$
Marissa: $2\left(4r\right)\ +\ 2\left(4s\right)$
Clara: $2\left(5r\ +\ 3r\right)\ +\ 4\left(2s\right)$
Alex: $24\ \times\ r\ \times\ s$

Part A
Which expressions are equivalent to the perimeter of the figure? List all of the correct expressions. Show how you know the correct expressions are equivalent to each other.

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

the first equation 2 x -4 = -8

the second equation -4 x 5 x -6 is 120

you should add the points so we know what to choose from.   but to figure out the answer, graph it.   then any points the are on the line are solutions.