Solve each problem. A cashier has a total of 30 bills, made...

Solve each problem.
A cashier has a total of 30 bills, made up of ones, fives, and twenties. The number of twenties is 9 more than the number of ones. The total value of the money is $\$ 351 .$ How many of each denomination of bill are there?

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Key Concepts

Diophantine Equations

Diophantine equations are algebraic equations that seek integer solutions. In problems that involve counting physical items like bills, solutions must be whole numbers, making the equations inherently Diophantine in nature and requiring techniques that ensure only integer answers are considered.

Substitution Method

The substitution method is a strategy for solving systems of equations by expressing one variable in terms of others and then substituting this expression into the remaining equations. This method simplifies the system, reducing the number of variables and making it easier to solve for each unknown sequentially.

Variables

Variables are symbols used to represent unknown quantities in a mathematical problem. Assigning variables to each denomination or component allows one to formulate equations based on the given conditions, facilitating a structured approach to finding the answer.

System of Linear Equations

A system of linear equations involves multiple linear relationships that share the same set of variables. In real-world problems, such as counting bills or coins, these equations help model constraints like total count and total value so that the solution meets all conditions simultaneously.

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