In a bag, a child has 325 coins worth 19.50 . There were...

In a bag, a child has 325 coins worth $\$ 19.50$ . There were three toins: pennies, nickels, and dimes. If the bag contained the same number of nickels as dimes, how many of each type of coin was in the bag?

Key Concepts

System of Linear Equations

This concept involves setting up and solving multiple equations simultaneously where each equation represents a different aspect of a problem. In coin problems, for instance, one equation might represent the total number of coins, while another represents the total monetary value. The system is solved to find the values of the unknowns that satisfy all given conditions.

Coin Value Equations

Coin value equations are used in problems involving currency where each type of coin has a specific monetary value. These equations relate the number of each type of coin to the total value. By multiplying the count of each coin by its corresponding value and summing these products, one obtains an equation that represents the total monetary amount.

Constraints and Equality Conditions

Many word problems include additional constraints, such as having equal numbers of certain items. In coin problems, a condition like having the same number of nickels and dimes adds an extra layer of restriction, reducing the number of unknowns. This equality condition is crucial as it allows one to substitute or combine variables, simplifying the system of equations.

Substitution Method

The substitution method is a common technique for solving systems of equations where one expresses one variable in terms of another using one of the equations, and then substitutes this expression into another equation. This method is particularly useful when the problem provides a direct relationship between some of the unknowns, such as having an equal number of two types of coins.

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