Solve each problem. See Examples 5 and 9. 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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1
Define variables: Let x be the number of one-dollar bills, y be the number of five-dollar bills, and z be the number of twenty-dollar bills.
Set up the first equation based on the total number of bills: x + y + z = 30.
Set up the second equation based on the relationship between ones and twenties: z = x + 9.
Set up the third equation based on the total value of the money: 1x + 5y + 20z = 351.
Substitute z = x + 9 into the first and third equations, then solve the system of equations to find the values of x, y, and z.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
System of Equations
A system of equations consists of two or more equations that share variables. To solve the problem, we can set up equations based on the relationships described: the total number of bills and the value of the bills. This allows us to find the values of the unknowns, which in this case are the quantities of each type of bill.
In algebra, variables represent unknown quantities, and expressions are combinations of variables and constants. For this problem, we can define variables for the number of ones, fives, and twenties. By expressing the relationships between these variables, we can create equations that reflect the conditions given in the problem.
Word problems require translating a real-world scenario into mathematical expressions and equations. This involves identifying key information, such as totals and relationships, and converting them into a solvable format. Understanding how to extract and represent this information is crucial for solving the problem effectively.