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]12x^2 - 2xy + 3[/tex]. The friends have already collected the following number of cans:

Jessa: [tex]3x^2[/tex]
Tyree: [tex]5x^2 - 8[/tex]
Ben: [tex]3xy + 4[/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 :

To solve this problem, we'll break it down into two parts: finding the total number of canned foods collected by the three friends and finding out how many more are needed to meet their goal.

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

Let's write down the number of cans each friend has collected:

- Jessa: [tex]\( 3x^2 \)[/tex]
- Tyree: [tex]\( 5x^2 - 8 \)[/tex]
- Ben: [tex]\( 3xy + 4 \)[/tex]

To find the total number of cans collected so far, we need to add these expressions together:

[tex]\[ \text{Total collected} = (3x^2) + (5x^2 - 8) + (3xy + 4) \][/tex]

Next, combine like terms:

[tex]\[ 3x^2 + 5x^2 + 3xy - 8 + 4 \][/tex]

Combine the [tex]\( x^2 \)[/tex] terms:

[tex]\[ (3x^2 + 5x^2) = 8x^2 \][/tex]

Combine the constant terms:

[tex]\[ -8 + 4 = -4 \][/tex]

Putting it all together:

[tex]\[ 8x^2 + 3xy - 4 \][/tex]

So, the expression representing the total amount of canned food collected so far by the three friends is:

[tex]\[ 8x^2 + 3xy - 4 \][/tex]

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

The goal for the canned food collection is given by:

[tex]\[ 12x^2 - 2xy + 3 \][/tex]

To find how many more cans are needed, we subtract the total collected amount from the goal amount:

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

Substitute the expressions we found:

[tex]\[ \text{Remaining to collect} = (12x^2 - 2xy + 3) - (8x^2 + 3xy - 4) \][/tex]

Distribute the subtraction:

[tex]\[ 12x^2 - 2xy + 3 - 8x^2 - 3xy + 4 \][/tex]

Combine like terms:

[tex]\[ (12x^2 - 8x^2) + (-2xy - 3xy) + (3 + 4) \][/tex]

Simplify:

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

So, the expression representing the number of cans the friends still need to collect to meet their goal is:

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

### Summary

Part A:
The expression for the total amount of canned food collected so far is:

[tex]\[ 8x^2 + 3xy - 4 \][/tex]

Part B:
The expression for the number of cans still needed to meet their goal is:

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