Sure! Let's tackle each part step-by-step.
### Part A:
We need to find an expression that represents the total amount of canned food collected so far by Jessa, Tyree, and Ben.
1. Jessa's collected cans:
[tex]\[ 7xy + 3 \][/tex]
2. Tyree's collected cans:
[tex]\[ 3x^2 - 4 \][/tex]
3. Ben's collected cans:
[tex]\[ 5x^2 \][/tex]
To find the total collected cans so far, we add these three expressions together.
[tex]\[
\text{Total collected} = (7xy + 3) + (3x^2 - 4) + (5x^2)
\][/tex]
Combine the like terms (terms with [tex]\(x^2\)[/tex], [tex]\(xy\)[/tex], and constants):
[tex]\[
= 7xy + 3x^2 - 4 + 5x^2 + 3
\][/tex]
Grouping the [tex]\(x^2\)[/tex] terms together:
[tex]\[
= 8x^2 + 7xy + (3 - 4)
\][/tex]
Simplify the constants:
[tex]\[
= 8x^2 + 7xy - 1
\][/tex]
Therefore, the expression representing the total amount of canned food collected so far is:
[tex]\[
8x^2 + 7xy - 1
\][/tex]
### Part B:
Next, we need to find an expression representing the number of cans the friends still need to collect to meet their goal.
We are given that the goal for their canned food collection is expressed as:
[tex]\[
10x^2 + 7xy + 8
\][/tex]
We will subtract the total number of cans they have already collected from the goal to find the remaining cans needed.
[tex]\[
\text{Cans needed} = (\text{Goal}) - (\text{Total collected})
\][/tex]
Substitute the expressions for the goal and the total collected:
[tex]\[
\text{Cans needed} = (10x^2 + 7xy + 8) - (8x^2 + 7xy - 1)
\][/tex]
Distribute the negative sign and combine like terms:
[tex]\[
= 10x^2 + 7xy + 8 - 8x^2 - 7xy + 1
\][/tex]
Group the [tex]\(x^2\)[/tex] terms and the constants:
[tex]\[
= (10x^2 - 8x^2) + (7xy - 7xy) + (8 + 1)
\][/tex]
Simplify the expression:
[tex]\[
= 2x^2 + 9
\][/tex]
Therefore, the expression representing the number of cans the friends still need to collect to meet their goal is:
[tex]\[
2x^2 + 9
\][/tex]
In summary:
- Part A: The total amount of canned food collected so far is given by [tex]\(8x^2 + 7xy - 1\)[/tex].
- Part B: The number of cans still needed to meet their goal is given by [tex]\(2x^2 + 9\)[/tex].
