Mr. Yi buys vegetables at a market. He purchases 6 pounds of potatoes, p, and 3 pounds of onions, n, for $18. Onions cost twice as much as potatoes. To determine the unit price for each item, his daughter sets up and solves the system of equations shown.
1. [tex]6p + 3n = 18[/tex]
2. [tex]n = 2p[/tex]
3. [tex]6(2n) + 3n = 18[/tex]
4. [tex]12n + 3n = 18[/tex]
5. [tex]15n = 18[/tex] [tex]n = $1.20[/tex]
Onions cost $1.20 per pound.
Analyze the daughter's solution. Which statements are true? Check all that apply.
A. The equation [tex]2n = p[/tex] should be [tex]2p = n[/tex].
B. The equation [tex]6p + 3n = 18[/tex] should be [tex]6n + 3p = 18[/tex].
C. The actual cost of the onions is $3.00 per pound.
D. Potatoes cost $0.60 per pound.
E. Potatoes cost $1.50 per pound.
F. Potatoes cost $2.40 per pound.
Answer (2)
Define variables p for the cost per pound of potatoes and n for the cost per pound of onions.
Establish the equations: 6 p + 3 n = 18 and n = 2 p .
Substitute n = 2 p into the first equation and solve for p , finding p = 1.50 .
Calculate the cost of onions using n = 2 p , resulting in n = 3.00 . The true statements are: The equation 2 n = p should be 2 p = n , The actual cost of the onions is $3.00 per pound, and Potatoes cost $1.50 per pound.
The final answer is: The equation 2 n = p should be 2 p = n , The actual cost of the onions is $3.00 per pound, and Potatoes cost $1.50 per pound.
Explanation
Problem Analysis Let's analyze the problem. Mr. Yi bought 6 pounds of potatoes and 3 pounds of onions for a total of $18. We also know that onions cost twice as much as potatoes. The goal is to find the correct cost per pound for both potatoes and onions and to evaluate the daughter's solution.
Defining Variables and Equations Let's define our variables:
p = cost per pound of potatoes
n = cost per pound of onions
From the problem statement, we can write two equations:
The total cost equation: 6 p + 3 n = 18
The relationship between the cost of onions and potatoes: n = 2 p (since onions cost twice as much as potatoes)
Solving for Potato Cost Now, let's substitute the second equation into the first equation to solve for the cost of potatoes:
6 p + 3 ( 2 p ) = 18
6 p + 6 p = 18
12 p = 18
p = 12 18 = 2 3 = 1.50
So, potatoes cost $1.50 per pound.
Solving for Onion Cost Now that we know the cost of potatoes, we can find the cost of onions:
n = 2 p = 2 ( 1.50 ) = 3.00
So, onions cost $3.00 per pound.
Evaluating the Statements Now, let's evaluate the statements:
The equation 2 n = p should be 2 p = n . (True)
The equation 6 p + 3 n = 18 should be 6 n + 3 p = 18 . (False)
The actual cost of the onions is $3.00 per pound. (True)
Potatoes cost $0.60 per pound. (False)
Potatoes cost $1.50 per pound. (True)
Potatoes cost $2.40 per pound. (False)
Final Answer Therefore, the true statements are:
The equation 2 n = p should be 2 p = n .
The actual cost of the onions is $3.00 per pound.
Potatoes cost $1.50 per pound.
Examples
Understanding systems of equations is very useful in everyday life. For example, imagine you're planning a balanced diet. You need a certain number of calories and a specific amount of protein each day. Different foods have different amounts of calories and protein. By setting up a system of equations, you can determine the right amounts of each food to meet your dietary needs. This is similar to how we found the cost of potatoes and onions, but instead of money, we're dealing with nutrients!
The true statements regarding Mr. Yi's vegetable purchase are that the equation should be 2 p = n , the cost of onions is $3.00 per pound, and potatoes cost $1.50 per pound.
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