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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Define variables for the number of each type of bill: 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.

Write an equation for the total number of bills: since there are 30 bills in total, we have $x + y + z = 30$.

Express the relationship between the number of twenties and ones: the number of twenties is 9 more than the number of ones, so $z = x + 9$.

Write an equation for the total value of the bills: the total amount is $351$, so $1 \cdot x + 5 \cdot y + 20 \cdot z = 351$.

Substitute $z = x + 9$ into the first and third equations to reduce the system to two equations with two variables, then solve for $x$ and $y$, and finally find $z$ using $z = x + 9$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Setting Up Variables for Word Problems

Assign variables to represent unknown quantities in the problem, such as the number of one-dollar, five-dollar, and twenty-dollar bills. This step translates the word problem into algebraic expressions, making it easier to form equations based on the given relationships.

Formulating Systems of Linear Equations

Use the relationships described in the problem to create multiple linear equations. For example, the total number of bills, the relationship between the number of twenties and ones, and the total value of the bills each provide an equation that together form a system to solve.

Solving Systems of Equations

Apply methods such as substitution or elimination to solve the system of linear equations. This process finds the values of the variables that satisfy all equations simultaneously, revealing the number of each type of bill.