Question

Sal's Sandwich Shop sells wraps and sandwiches as part of its lunch specials. The profit on every sandwich is $2, and the profit on every wrap is $3. Sal made a profit of $1,470 from lunch specials last month. The equation 2x + 3y = 1,470 represents Sal's profits last month, where x is the number of sandwich lunch specials sold and y is the number of wrap lunch specials sold.

Change the equation to slope-intercept form. Identify the slope and y-intercept of the equation. Be sure to show all your work.
Describe how you would graph this line using the slope-intercept method. Be sure to write using complete sentences.
Write the equation in function notation. Explain what the graph of the function represents. Be sure to use complete sentences.
Graph the function. On the graph, make sure to label the intercepts. You may graph your equation by hand on a piece of paper and scan your work or you may use graphing technology.
Suppose Sal's total profit on lunch specials for the next month is $1,593. The profit amounts are the same: $2 for each sandwich and $3 for each wrap. In a paragraph of at least three complete sentences, explain how the graphs of the functions for the two months are similar and how they are different.

Answer 1

That's a whole bunch of questions you got there mhmm

But let's go through them one at a time

1. Change the equation to slope-intercept form. Identify the slope and y-intercept of the equation. Be sure to show all your work.

Slope-intercept form is
y = mx + b

where m is the slope
and b is the y-intercept

Answer 2

The equation they gave you is:
2x + 3y = 1,470

Can you solve the equation for y?
@alexis1415

If you can't, I can help you solve for y

Answer 3

i kind of forgot how to solve for y

Answer 4

No worries!

When we say "solve for y"

what we mean is: get y all by itself on one side of the equal sign and everything else on the other side of the equal sign

Answer 5

So we have
2x + 3y = 1,470

Let's move that 2x to the other side

Since we're ADDING 2x to 3y
We need to do the OPPOSITE on both sides

What is the opposite of addition?
Is it subtraction? Multiplication? Division?

Answer 6

subtraction

Answer 7

Good! So we have to subtract 2x on BOTH sides

so

2x + 3y - 2x = 1470 - 2x

what is
2x - 2x =

Answer 8

0

Answer 9

Good! So now we have on the left side

3y = 1470 - 2x

another way to write that 1470 - 2x

1470 + (-2x)
and then since they're both being added, we can switch it around
because think about it like 1 + 2 is the same as 2 + 1

so we would get
-2x + 1470

so our equation now is

3y = -2x + 1470

Answer 10

Does that make sense?

Answer 11

yes

Answer 12

1470 - 2x

is just the same thing as 1470 + (-2x)

when you're adding a negative number, it's basically subtraction

Answer 13

okay good! so now your equation is starting to look more and more like y = mx + b

but we have one last thing
we have 3y
we want it to just be 'y' on the left side

so do we have to multiply or divide 3 on both sides to get rid of the 3 with the y?

Answer 14

remember
3y means 3 multiplied by y

so we have to do the opposite of that on both sides to get rid of the 3

Answer 15

divide

Answer 16

Good! So let's divide 3 and we get

\( y = \dfrac{-2x + 1470}{3}\)

Answer 17

can you divide each one by 3

remember

\(\dfrac{a + b}{c} = \dfrac{a}{c} + \dfrac{b}{c}\)

Answer 18

-2x/3 will stay the same because we can't simplify it anymore

but what is 1470 / 3

Answer 19

1470 / 3 = 490

Answer 20

Good! So now we have our equation finally in slope-intercept form

\( y = -\dfrac{2}{3}x + 490\)

Answer 21

#2 Describe how you would graph this line using the slope-intercept method. Be sure to write using complete sentences.

Do you know what a y-intercept is? And what a slope is?

Answer 22

so we don't go over those anymore and i forgot

Answer 23

That's okay

The y-intercept is when the line crosses the y-axis. That's when x = 0

So remember how the equation is written in slope-intercept form
y = mx + b

b is the y-intercept
That means that one of the points of our graph is going to be (0, b)

Answer 24

so if your equation is

\( y = -\dfrac{2}{3}x + 490\)

what is the y-intercept?

Answer 25

(0,490)

Answer 26

Excellent!!

Now once more
y = mx + b

the slope is 'm'

and slope is rise / run

it means that from one point, you will rise some number and then run some number

here's an image that might help you (it's with different numbers so don't get confused)

Answer 27

So for our equation, what is the slope?

Answer 28

it would be up 2 right 3 or the actual equation

Answer 29

so you're almost right! But remember the slope is NEGATIVE 2 / 3


So since we have -2 /3
and it's RISE over RUN

that means we rise by -2
but if we're rising by a negative number that means we're going DOWN not up
and then we're running 3 to the right

Answer 30

oh ok

Answer 31

yeah because, i forgot that it was negative but if i remembered it i would have been able to answer that

Answer 32

so we go down by -2 and to the right by 3

we can also go UP by positive 2 but then we would be running by NEGATIVE 3 so that means we're moving towards the left

that's how you get the two different directions

Answer 33

does that make sense?

for the first time, we kept the negative with 2
but in the second time, we put the negative with 3

Answer 34

yes it makes sense

Answer 35

Good! Ready to move on to the second question?

Answer 36

yes

Answer 37

err third*

3) Write the equation in function notation. Explain what the graph of the function represents. Be sure to use complete sentences.

when they say function notation, they basically mean
replace the `y` with f(x)

Answer 38

oh ok, yeah i was wondering how sometimes that would be on there, thank you

Answer 39

yup! f(x) is just the same thing as y

except that instead of calling it an equation we'll be calling it a function

Answer 40

The second half of that question says "explain what the graph of the function represents"

They tell you that in your question already:
`The equation 2x + 3y = 1,470 represents Sal's profits last month, where x is the number of sandwich lunch specials sold and y is the number of wrap lunch specials sold.`

Answer 41

oh ok, so i write where it goes on the graph by doing what we did not to long ago

Answer 42

Yes, you can just explain that on our graph, the x-values is the number of sandwiches sold and y is the number of wrap sandwiches sold so that way you have a profit of 1470

Answer 43

ok

Answer 44

so i think i get it now actually :)

Answer 45

Wonderful! :D

Answer 46

So next part:

4) Graph the function. On the graph, make sure to label the intercepts. You may graph your equation by hand on a piece of paper and scan your work or you may use graphing technology.

Answer 47

graphing by hand is too much work esp using the drawing tool, but you you know how to graph it?

Answer 48

Alternatively, they do permit the usage of graphing technology so may I suggest https://www.desmos.com/calculator

Answer 49

oh well thank you, I will use that

Answer 50

Tell me when you're done with that so we can move on to the next part!

Answer 51

I'm ready :)

Answer 52

Wonderful! :D

So now we have the last question

5) Suppose Sal's total profit on lunch specials for the next month is $1,593. The profit amounts are the same: $2 for each sandwich and $3 for each wrap. In a paragraph of at least three complete sentences, explain how the graphs of the functions for the two months are similar and how they are different.

Answer 53

So remember the original equation they gave us?

2x + 3y = 1470

That 1470 was the total profit they made

But now this question is saying that the total profit would be 1593

Can you write the equation for that?
(Basically, you just replace 1470 with 1593)

Answer 54

so 2x + 3y = 1593

Answer 55

There you go!

And let's solve it for `y` too, shall we?

It'll make it easier for you to put in the graphing calculator

Remember to move the 2x to the other side and then divide by 3 on both sides.

What do you end up with?

Answer 56

Well actually we don't even need to bother

Answer 57

oh ok

Answer 58

So how are the graphs similar? And how are they different?

https://www.desmos.com/calculator/er0f1aibkh

Answer 59

Here's the graph:

Answer 60

they are similar because, they are both going in the same direction but with different points

Answer 61

they also both go through the x and y axis

Answer 62

Yes! They are parallel lines

Answer 63

yes i didn't know what the word was and it was on the tip of my tounge thank you

Answer 64

No problem! Glad I could help :)

Answer 65

i appreciate all of your help with all of these questions, it's my first time being on this app and thank you!

Answer 66

Of course! It was my pleasure!!

Welcome to QuestionCove!!

(Also if it's too difficult to respond on the app, you can always open it up on your laptop/computer: https://questioncove.com)

Answer 67

alright thank you! have nice day