Problem 29 - Algebra 1: Concepts and Skills

Question

Match the situation with the corresponding linear system. You have only 1 dollar bills and 5 dollar bills in your wallet. There are 7 bills worth a total of 19 dollar. $$ A. x+y=7 x+3 y=19$$ $$B. x+y=7 x+5 y=19$$ $$C. x+y=5 3 x+5 y=19$$

Step-by-Step Solution

Answer

The correct linear system that matches the given situation is option B: x + y = 7 and x + 5y = 19.

Step 1: Identify the Variables

Let's start by interpreting the problem. We know that there are only two types of bills, one-dollar bills and five-dollar bills. We can denote one-dollar bills as \( x \) and five-dollar bills as \( y \). The problem states that there are 7 bills in total which are worth 19 dollars.

Step 2: Interpret the Equations

The setup given implies two equations. One is the total number of bills (x+y=7), implying that the number of one dollar bills (x) and the number of five dollar bills (y) add to seven. The other is the total value of the bills (x+5y=19), meaning the total sum resulting from the number of one dollar bills plus five times the number of five dollar bills equals 19 dollars.

Step 3: Match with Given Options

Now we can match those equations to the given options. The correct system of equations will have two equations: one equation showing that x+y=7 and another showing x+5y=19. The only option that matches this pair of equations is B. x+y=7, x+5y=19. This is the linear system of equations that represent the situation in the problem.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that set two algebraic expressions equal to each other. In this exercise, linear equations are essential for modeling real-world situations using math. These equations often include variables whose values we are solving for.
An example from the problem is the equation that represents the total number of bills:

• \( x + y = 7 \)

This equation derives from the information that there are 7 total bills in a wallet. Linear equations like this form the backbone of many mathematical models because they are simple yet powerful tools for understanding relationships between variables.

Variables

Variables in linear equations function as placeholders for unknown values that need to be found. In our exercise, we have two variables: \(x\) and \(y\). These represent the number of one-dollar and five-dollar bills, respectively.
Understanding what each variable stands for is crucial:

• \( x \) indicates the count of one-dollar bills.
• \( y \) indicates the count of five-dollar bills.

By introducing variables, we translate the real-world situation into a format that can be solved using mathematics. This transformation is the first step in solving any linear equation.

Equation Solving

Equation solving involves finding the values of the variables that satisfy the given equations. For a system of linear equations like in our exercise, this process is about finding common values of \(x\) and \(y\) that satisfy all the given equations simultaneously.
In our problem, we solved:

• \( x + y = 7 \)
• \( x + 5y = 19 \)

To solve these equations, one can use different strategies such as substitution or elimination. These steps help simplify the equations, allowing us to find specific values for the variables that meet all conditions set by the equations.

Mathematical Modeling

Mathematical modeling is about creating a set of equations to represent a real-world situation. In the exercise, we modeled a scenario using two linear equations. These equations capture the essential relationships: the number of bills and their total value.
By translating the "total amount of bills" and "total sum of money" scenarios into equations, we used mathematical modeling to describe and solve the situation:

• The total number of bills equation: \( x + y = 7 \)
• The total dollar value equation: \( x + 5y = 19 \)

Modeling helps us understand and predict outcomes in various situations, making abstract problems tangible and solvable with mathematical tools.