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Which system of equations best represents the situation below?

A farmer grew his own tomatoes (a), eggplants (b), and potatoes (c). He decided to package his vegetables and price them as follows:
- 2 tomatoes, 1 eggplant, 3 potatoes for \$10
- 3 tomatoes, 2 eggplants, 4 potatoes for \$14
- 4 tomatoes, 2 eggplants, 6 potatoes for \$20

A.
[tex]\[2a + b + 3c = 10\][/tex]
[tex]\[3a + 2b + 4c = 14\][/tex]
[tex]\[4a + 2b + 6c = 20\][/tex]

B.
[tex]\[2a + b + 3c = 10\][/tex]
[tex]\[3a + 2b + 4c = 14\][/tex]
[tex]\[2a + 3b + 6c = 20\][/tex]

C.
[tex]\[2a + b + 3c = 10\][/tex]
[tex]\[3a + 2b + 4c = 14\][/tex]
[tex]\[4a + 2b + 6c = 20\][/tex]

D.
[tex]\[\begin{array}{l}
a + 2b + 4c = 14 \\
2a + 3b + 6c = 20
\end{array}\][/tex]



Answer :

GinnyAnswer
To identify which system of equations best represents the given equations, we start by analyzing each provided option and comparing it to the original set of equations:

### Original Equations
1. \(2a + b + 3c = 10\)
2. \(3a + 2b + 4c = 14\)
3. \(4a + 2b + 6c = 20\)

### Option A
1. \(3a + 2b + 4c = 14\)
2. \(4a + 2b + 6c = 20\)
3. \(2a + 2b + 3c = 10\)

### Option B
1. \(3a + 2b + 4c = 14\)
2. \(2a + 3b + 6c = 20\)
3. \(2a + 2b + 3c = 10\)

### Option C
1. \(3a + 2b + 4c = 14\)
2. \(4a + 2b + 6c = 20\)
3. \(2a + 2b + 2c = 10\)

### Option D
1. \(a + 2b + 4c = 14\)
2. \(2a + 3b + 6c = 20\)


Now, let's compare each option with the original equations:

- Option A:
- First equation \(3a + 2b + 4c = 14\) matches the second original equation.
- Second equation \(4a + 2b + 6c = 20\) matches the third original equation.
- Third equation \(2a + 2b + 3c = 10\) does not match any of the original equations.

- Option B:
- First equation \(3a + 2b + 4c = 14\) matches the second original equation.
- Second equation \(2a + 3b + 6c = 20\) does not match any of the original equations.
- Third equation \(2a + 2b + 3c = 10\) does not match any of the original equations.

- Option C:
- First equation \(3a + 2b + 4c = 14\) matches the second original equation.
- Second equation \(4a + 2b + 6c = 20\) matches the third original equation.
- Third equation \(2a + 2b + 2c = 10\) does not match any of the original equations.

- Option D:
- First equation \(a + 2b + 4c = 14\) does not match any of the original equations.
- Second equation \(2a + 3b + 6c = 20\) does not match any of the original equations.

### Conclusion
Out of all the options provided, Option C [ \(3a + 2b + 4c = 14\), \(4a + 2b + 6c = 20\), and \(2a + 2b + 2c = 10\) ] best aligns with two out of the three original equations. Thus, the correct choice is:

Option C

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