To determine the correct system of equations that models the situation where Anthony has a cup full of nickels and dimes worth [tex]$6.80 and there are 82 coins in total, let's break down the information given:
1. We need to use \( x \) to represent the number of nickels and \( y \) to represent the number of dimes.
2. Each nickel is worth 5 cents and each dime is worth 10 cents.
3. The total value of the nickels and dimes is $[/tex]6.80, which is equivalent to 680 cents.
4. The total number of coins (nickels and dimes) is 82.
Based on this information, we can form two equations:
1. The total value equation: [tex]\( 5x + 10y = 680 \)[/tex]
2. The total number of coins equation: [tex]\( x + y = 82 \)[/tex]
Let's examine each option to determine which one correctly represents these equations:
A. [tex]\(\left\{\begin{array}{l}x+y=680 \\ 5x+10y=82\end{array}\right.\)[/tex]
- This option is incorrect because it places the total number of coins equal to 680, which should instead be 82, and the total value equal to 82, which should instead be 680.
B. [tex]\(\left\{\begin{array}{l}5x+10y=680 \\ x+y=82\end{array}\right.\)[/tex]
- This option is correct because it accurately states the total value equation ([tex]\(5x + 10y = 680\)[/tex]) and the total number of coins equation ([tex]\(x + y = 82\)[/tex]).
C. [tex]\(\left\{\begin{array}{l}5x+y=680 \\ x+10y=82\end{array}\right.\)[/tex]
- This option is incorrect because it mixes up the coefficients and the constants. It suggests a different relationship which does not represent the given information.
D. [tex]\(\left\{\begin{array}{l}5x+y=82 \\ x+10y=680\end{array}\right.\)[/tex]
- This option is incorrect because, like option C, it incorrectly mixes up the relationships between the variables.
Therefore, the correct system of equations that models the given situation is:
[tex]\[ \left\{\begin{array}{l}
5x + 10y = 680 \\
x + y = 82
\end{array}\right. \][/tex]
And this corresponds to option B.
