Question

The produce market is having a sale on grapes and oranges. A customer found the following two receipts in her cart as she was walking into the store. Use the information from the receipts to write a system of linear equations that can be used to determine the price per pound of grapes and the price per pound of oranges. Then, use the system to test the following solutions to determine which solution is viable. A. Given these constraints, grapes cost $3.25 per pound, and oranges cost $1.05 per pound. B. Given these constraints, grapes cost $1.60 per pound, and oranges cost $2.15 per pound. C. Given these constraints, grapes cost $1.05 per pound, and oranges cost $3.25 per pound. D. Give these constraints, grapes cost $2.15 per pound, and oranges cost $1.60 per pound.

Answer 1

Step-by-step explanation:

From the first receipt:

2 pounds of grapes + 4 pounds of oranges = 10.70 which, in an algebraic equation, looks like this:

2g + 4o = 10.70

From the second receipt:

3 pounds of grapes + 2 pounds of oranges = 9.65 which, in an algebraic equation, looks like this:

3g + 2o = 9.65

Putting those together into a system and solving using the elimination method:

2g + 4o = 10.7

3g + 2o = 9.65

I am going to eliminate the oranges first since it's easier to do that. I will multiply the second equation by -2 to get a new system:

2g + 4o = 10.7

-6g - 4o = -19.3

As you can see, the oranges are eliminated because 4o - 4o = 0o. That leaves us with only the grapes:

-4g = -8.6 so

g = 2.15

Grapes cost $2.15 per pound. Now sub that into either one of the original equations to solve for the cost per pound of oranges:

2(2.15) + 4o = 10.7 and

4.3 + 4o = 10.7 and

4o = 6.4 so

o = 1.60

Oranges cost $1.60 per pound. That is choice D from your list.

Answer 2

None of the provided solutions (A, B, C, D) match both equations derived from the receipts simultaneously, indicating none are viable given the constraints.

Step-by-step explanation:

Let's denote the price per pound of grapes as G and the price per pound of oranges as O.

From the receipts, we have two pieces of information:

Receipt 1:

3 pounds of grapes and 4 pounds of oranges for $13.75

This can be written as the equation:

3G + 4O = 13.75

Receipt 2:

5 pounds of grapes and 2 pounds of oranges for $14.75

This can be written as the equation:

5G + 2O = 14.75

So, the system of equations representing the given information is:

`\[\begin{cases} 3G + 4O = 13.75 \\ 5G + 2O = 14.75 \end{cases}\]`

Now, let's test the given solutions:

A. G = 3.25 and O = 1.05

Plugging these values into the equations:

`\[3 * 3.25 + 4 * 1.05 = 9.75 + 4.20 = 13.95 \\eq 13.75\]`

`\[5 * 3.25 + 2 * 1.05 = 16.25 + 2.10 = 18.35 \\eq 14.75\]`

B. G = 1.60 and O = 2.15

Plugging these values into the equations:

`\[3 * 1.60 + 4 * 2.15 = 4.80 + 8.60 = 13.40 \\eq 13.75\]`

`\[5 * 1.60 + 2 * 2.15 = 8.00 + 4.30 = 12.30 \\eq 14.75\]`

C. G = 1.05 and O = 3.25

Plugging these values into the equations:

`\[3 * 1.05 + 4 * 3.25 = 3.15 + 13.00 = 16.15 \\eq 13.75\]`

`\[5 * 1.05 + 2 * 3.25 = 5.25 + 6.50 = 11.75 \\eq 14.75\]`

D. G = 2.15 and O = 1.60

Plugging these values into the equations:

`\[3 * 2.15 + 4 * 1.60 = 6.45 + 6.40 = 12.85 \\eq 13.75\]`

`\[5 * 2.15 + 2 * 1.60 = 10.75 + 3.20 = 13.95 \\eq 14.75\]`

None of the given solutions match both equations simultaneously. Therefore, none of the provided solutions satisfy the given constraints.