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Question 1 (Essay Worth 10 points)
[tex]$(06.02 MC )$[/tex]

Three friends, Jessa, Tyree, and Ben, are collecting canned food for a culinary skills class. Their canned food collection goal is represented by the expression [tex]$10 x^2 - 4 x y + 12$[/tex].

The friends have already collected the following number of cans:

- Jessa: [tex]$7 x y + 3$[/tex]
- Tyree: [tex]$3 x^2 - 4$[/tex]
- Ben: [tex]$5 x^2$[/tex]

Part A: Write an expression to represent the amount of canned food collected so far by the three friends. Show all your work. (5 points)

Part B: Write an expression that represents the number of cans the friends still need to collect to meet their goal. Show all your work. (5 points)



Answer :

GinnyAnswer
To address this problem, we need to calculate the number of cans collected so far and determine how many more cans are needed to reach the goal. Let's break this down step-by-step.

### Part A: Expression for the Amount of Canned Food Collected So Far
First, let's identify the number of cans each friend has collected:

- Jessa has collected [tex]\( 7xy + 3 \)[/tex] cans.
- Tyree has collected [tex]\( 3x^2 - 4 \)[/tex] cans.
- Ben has collected [tex]\( 5x^2 \)[/tex] cans.

To find the total number of cans collected so far, we sum up all these expressions:

[tex]\[ \text{Total collected} = (7xy + 3) + (3x^2 - 4) + (5x^2) \][/tex]

Next, we combine like terms:

- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( 3x^2 + 5x^2 = 8x^2 \)[/tex]
- Combine the [tex]\( xy \)[/tex] terms: There's only one [tex]\( xy \)[/tex] term, [tex]\( 7xy \)[/tex]
- Combine the constant terms: [tex]\( 3 - 4 = -1 \)[/tex]

So, the total collected can be expressed as:

[tex]\[ \text{Total collected} = 8x^2 + 7xy - 1 \][/tex]

This represents the amount of canned food collected so far.

### Part B: Expression for the Number of Cans Still Needed to Meet Their Goal
The friends’ goal is given by the expression [tex]\( 10x^2 - 4xy + 12 \)[/tex].

To determine the number of cans still needed, we need to subtract the amount collected from the goal. This can be set up as follows:

[tex]\[ \text{Remaining needed} = (\text{Goal expression}) - (\text{Total collected}) \][/tex]

Substitute the expressions:

[tex]\[ \text{Remaining needed} = (10x^2 - 4xy + 12) - (8x^2 + 7xy - 1) \][/tex]

Now, let's simplify by distributing the negative sign and combining like terms:

[tex]\[ \text{Remaining needed} = 10x^2 - 4xy + 12 - 8x^2 - 7xy + 1 \][/tex]

Combine the like terms:

- Combine the [tex]\( x^2 \)[/tex] terms: [tex]\( 10x^2 - 8x^2 = 2x^2 \)[/tex]
- Combine the [tex]\( xy \)[/tex] terms: [tex]\( -4xy - 7xy = -11xy \)[/tex]
- Combine the constant terms: [tex]\( 12 + 1 = 13 \)[/tex]

So, the remaining cans needed can be expressed as:

[tex]\[ \text{Remaining needed} = 2x^2 - 11xy + 13 \][/tex]

### Summary:
- Part A: The total collected so far is [tex]\( 8x^2 + 7xy - 1 \)[/tex].
- Part B: The number of cans still needed to meet their goal is [tex]\( 2x^2 - 11xy + 13 \)[/tex].