Question

Marina volunteers at the Salvation Army. She has been tasked with buying nonperishable items for families that were displaced by a recent flood. She finds a company willing to sell her cans of food at a discounted price of $2.25 for a small can and $4.75 for a large can. She spent a total of $714.75 and bought 181 cans. Let x represent the number of small cans she purchased and y represent the number of large cans she purchased.

1) Write a system of equations for the situation
A) 2.25x+4.75y=1812.25x+4.75y=181
B) x+y=181x+y=181
C) x+y=714.75x+y=714.75
D) 2.25x+4.75y=714.752.25x+4.75y=714.75
2) Based on this context, are there any constraints on the variables that we should consider?
A) non-negative integers greater than 2.25 and less than 4.75
B) non-negative integers greater than 0 and less than 181
C) no constraints are needed on the variables.
D) integers greater than 0
3) Solve the system. How many small and large cans did Marina purchase?
Marina purchased __________ small cans and ____________ large cans.

Answer 1

Marina purchased 80 small cans and 0 large cans.

Step-by-step explanation:

The correct answer to the first question is option A) 2.25x+4.75y=181. This equation represents the total cost of the cans Marina bought, with x representing the number of small cans and y representing the number of large cans.

The correct answer to the second question is option B) non-negative integers greater than 0 and less than 181. Since Marina cannot buy a negative number of cans and the total number of cans she bought cannot exceed 181, these are the constraints on the variables.

To solve the system of equations, you can use substitution or elimination. Let's use elimination:

Multiply the first equation by 2.25 and the second equation by -2.25:

2.25(2.25x+4.75y) = 2.25(181)

-2.25(x+y) = -2.25(181)

Simplify:

5.0625x+10.6875y = 407.25

-2.25x-2.25y = -407.25

Add the two equations:

5.0625x+10.6875y -2.25x-2.25y = 407.25 -407.25 = 0

2.8125y = 0

Divide both sides by 2.8125:

y = 0

Substitute y = 0 into the first equation:

2.25x+4.75(0) = 181

2.25x = 181

Divide both sides by 2.25:

x = 80.44

Since x represents the number of small cans, it cannot be a decimal. Therefore, let's round x down to the nearest whole number:

x = 80

So, Marina purchased 80 small cans and 0 large cans.

Answer 2

1. The correct system of equations for the situation is: 2.25x + 4.75y = 181. The answer is A
2. Based on this context, the constraints on the variables that we should consider are non-negative integers greater than 0 and less than 181. The answer is B
3. Marina purchased 58 small cans and 123 large cans.

Step-by-step explanation:

1) In this system, x represents the number of small cans Marina purchased and y represents the number of large cans she purchased.

The equation 2.25x + 4.75y = 181 represents the total cost of the cans she bought, which should equal the total amount she spent. The equation accounts for the cost of the small cans (2.25x) and the cost of the large cans (4.75y), which should sum up to 181, the total amount she spent.

Therefore, the correct answer is A) 2.25x + 4.75y = 181.

2) Based on the context provided, there are constraints on the variables x and y that we should consider.

Since Marina is buying cans of food, it is reasonable to assume that the number of small cans (represented by x) and the number of large cans (represented by y) should be non-negative integers. We cannot have a negative number of cans.

Additionally, we need to consider the total number of cans Marina purchased. The problem states that she bought a total of 181 cans. Therefore, the sum of the number of small cans (x) and the number of large cans (y) should be equal to 181.

Taking these constraints into account, the correct answer is B) non-negative integers greater than 0 and less than 181.

So, the answer is A

3) To solve the system of equations, we can use the information given in the question. We have two variables, x and y, representing the number of small and large cans Marina purchased, respectively.

From the given information, we know that the cost of a small can is $2.25 and the cost of a large can is $4.75. Marina spent a total of $714.75 and bought 181 cans. We can express this information using the following equations:

• Equation 1: 2.25x + 4.75y = 714.75 (represents the total cost)
• Equation 2: x + y = 181 (represents the total number of cans)

To solve this system of equations, we can use substitution or elimination. Let's use substitution in this case.

Rearranging Equation 2, we get:

x = 181 - y

Substituting this value of x into Equation 1, we have:

2.25(181 - y) + 4.75y = 714.75

Expanding and simplifying, we get:

407.25 - 2.25y + 4.75y = 714.75

Combining like terms, we have:

2.5y = 307.5

Dividing both sides by 2.5, we get:

y = 123

Substituting this value of y into Equation 2, we can solve for x:

x + 123 = 181

x = 58

Therefore, Marina purchased 58 small cans and 123 large cans.