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Matthew has a jar of $368$ nickels and dimes he has been collecting. The total value of the coins is $\$28.40$ . Which system of linear equations and solutions can be used to represent the number of nickels and dimes Matthew has in his collection?

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Introduction to probability

Full solution

Q. Matthew has a jar of

368

nickels and dimes he has been collecting. The total value of the coins is

\$28.40

. Which system of linear equations and solutions can be used to represent the number of nickels and dimes Matthew has in his collection?

Equations Setup: Let's denote the number of nickels as $N$ and the number of dimes as $D$ . We know that each nickel is worth $\$0.05$ and each dime is worth $\$0.10$ . We can set up the following system of linear equations based on the information given: $\newline$ $1$ . The total number of coins is $368$ , which gives us the equation $N + D = 368$ . $\newline$ $2$ . The total value of the coins is $\$28.40$ , which gives us the equation $0.05N + 0.10D = 28.40$ .
Solve for N: First, we'll solve the first equation for one of the variables. Let's solve for N: $\newline$ $N = 368 - D$
Substitute $N$ into Equation: Next, we'll substitute the expression for $N$ into the second equation: $\newline$ $0.05(368 - D) + 0.10D = 28.40$ $\newline$ Now, we'll distribute the $0.05$ into the parentheses: $\newline$ $18.4 - 0.05D + 0.10D = 28.40$
Combine Like Terms: Now, we'll combine like terms: $\newline$ $18.4 + 0.05D = 28.40$ $\newline$ Next, we'll subtract $18.4$ from both sides to isolate the term with $D$ : $\newline$ $0.05D = 28.40 - 18.4$ $\newline$ $0.05D = 10.00$
Isolate D: Now, we'll divide both sides by $0.05$ to solve for $D$ : $\newline$ $D = \frac{10.00}{0.05}$ $\newline$ $D = 200$
Solve for D: Now that we have the value for D, we can substitute it back into the first equation to find N: $\newline$ $N = 368 - D$ $\newline$ $N = 368 - 200$ $\newline$ $N = 168$

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