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.
