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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]\(10x^2 - 4xy + 12\)[/tex]. The friends have already collected the following number of cans:

- Jessa: [tex]\(7xy + 3\)[/tex]
- Tyree: [tex]\(3x^2 - 4\)[/tex]
- Ben: [tex]\(5x^2\)[/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 tackle this question, let's break it down into two parts as specified.

### Part A: Expression for the Amount of Canned Food Collected So Far

Firstly, we need to find the amount of canned food each friend has collected.

1. Jessa's contribution:
[tex]\(7xy + 3\)[/tex]

2. Tyree's contribution:
[tex]\(3x^2 - 4\)[/tex]

3. Ben's contribution:
[tex]\(5x^2\)[/tex]

Next, we sum these individual contributions to find the total amount collected so far:

[tex]\[ (7xy + 3) + (3x^2 - 4) + (5x^2) \][/tex]

We then combine like terms in the expression:
- Combine the [tex]\(x^2\)[/tex] terms: [tex]\(3x^2 + 5x^2 = 8x^2\)[/tex]
- Combine the [tex]\(xy\)[/tex] terms: [tex]\(7xy\)[/tex] (no other [tex]\(xy\)[/tex] terms to combine with)
- Combine the constant terms: [tex]\(3 - 4 = -1\)[/tex]

Therefore, the total amount of canned food collected so far by the three friends is:

[tex]\[ 8x^2 + 7xy - 1 \][/tex]

### Part B: Expression for the Number of Cans Still Needed to Meet the Goal

The collection goal is expressed as:
[tex]\[ 10x^2 - 4xy + 12 \][/tex]

We have already calculated the total amount collected so far as:
[tex]\[ 8x^2 + 7xy - 1 \][/tex]

To find the number of cans still needed, we subtract the total collected amount from the goal. Thus, the expression for the number of additional cans required is:

[tex]\[ (10x^2 - 4xy + 12) - (8x^2 + 7xy - 1) \][/tex]

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

[tex]\[ 10x^2 - 4xy + 12 - 8x^2 - 7xy + 1 \][/tex]

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]

Therefore, the expression that represents the number of cans the friends still need to collect is:

[tex]\[ 2x^2 - 11xy + 13 \][/tex]

### Summary
- Part A: The quantity of canned food collected so far by the three friends is: [tex]\( 8x^2 + 7xy - 1 \)[/tex]
- Part B: The number of additional cans needed to meet their goal is: [tex]\( 2x^2 - 11xy + 13 \)[/tex]

This completes our detailed solution for both parts of the question.

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