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

Jessa: [tex]\(3x^2\)[/tex]

Tyree: [tex]\(5x^2 - 8\)[/tex]

Ben: [tex]\(2xy + 4\)[/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 :

GinnyAnswer
Certainly! Let's solve this problem step by step.

### Part A

First, let's write down the expressions for the number of cans each friend collected:

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

To find the total number of cans collected so far, we need to sum these three expressions.

The expression for the total collected cans will be:
[tex]\[ 3x^2 + (5x^2 - 8) + (2xy + 4) \][/tex]

Next, let's simplify this expression by combining like terms:
[tex]\[ 3x^2 + 5x^2 - 8 + 2xy + 4 \][/tex]

Now, sum the terms:
- Combine [tex]\(3x^2\)[/tex] and [tex]\(5x^2\)[/tex]:
[tex]\[ 3x^2 + 5x^2 = 8x^2 \][/tex]
- Combine the constant terms:
[tex]\[ -8 + 4 = -4 \][/tex]
- The [tex]\(2xy\)[/tex] term remains as is because there are no other like terms to combine with it.

Thus, the simplified expression for the total amount of canned food collected so far is:
[tex]\[ 8x^2 + 2xy - 4 \][/tex]

### Part B

The goal for the canned food collection is given by the expression:
[tex]\[ 12x^2 - 2xy + 3 \][/tex]

We need to find out how much more is required to meet this goal. This means we need to subtract the total collected so far from the goal. So we have to subtract the expression we got in Part A from the goal expression.

The expression for the remaining cans needed will be:
[tex]\[ (12x^2 - 2xy + 3) - (8x^2 + 2xy - 4) \][/tex]

Next, let's simplify this expression by distributing the negative sign:
[tex]\[ 12x^2 - 2xy + 3 - 8x^2 - 2xy + 4 \][/tex]

Now, combine like terms:
- Combine [tex]\(12x^2\)[/tex] and [tex]\(-8x^2\)[/tex]:
[tex]\[ 12x^2 - 8x^2 = 4x^2 \][/tex]
- Combine [tex]\(-2xy\)[/tex] and [tex]\(-2xy\)[/tex]:
[tex]\[ -2xy - 2xy = -4xy \][/tex]
- Combine [tex]\(3\)[/tex] and [tex]\(4\)[/tex]:
[tex]\[ 3 + 4 = 7 \][/tex]

Thus, the simplified expression for the number of cans the friends still need to collect is:
[tex]\[ 4x^2 - 4xy + 7 \][/tex]

### Summary

1. Part A: The expression representing the amount of canned food collected so far by the three friends is:
[tex]\[ 8x^2 + 2xy - 4 \][/tex]

2. Part B: The expression representing the number of cans the friends still need to collect to meet their goal is:
[tex]\[ 4x^2 - 4xy + 7 \][/tex]

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