Problem 28 - Algebra 2

Question

Marcus and Cody purchased game cards to play virtual games at the arcade. Marcus used 47 points from his game card to drive the race car simulator and the snowboard simulator four times each. Cody used 48.25 points from his game card to drive the race car five times and the snowboard three times. How many points does each game charge per play?

Step-by-Step Solution

Answer

Race car: 6.5 points per play, Snowboard: 5.25 points per play.

Step 1: Define Variables

Let \( x \) represent the points required for the race car simulator per play, and \( y \) represent the points needed for the snowboard simulator per play.

Step 2: Write Equations Based on Marcus

Based on Marcus's usage, we can write the equation: \[ 4x + 4y = 47 \] This equation represents his usage of both simulators four times each.

Step 3: Write Equations Based on Cody

Based on Cody's usage, we formulate the equation: \[ 5x + 3y = 48.25 \] This is based on driving the race car five times and the snowboard three times.

Step 4: Simplify Marcus's Equation

Since both \( x \) and \( y \) are used 4 times each by Marcus, we can simplify \[ 4(x + y) = 47 \] to \[ x + y = 11.75 \] This equation helps in making the substitution easier.

Step 5: Solve the Equations Simultaneously

Using substitution, substitute \( x = 11.75 - y \) from Marcus's equation into Cody's equation: \[ 5(11.75 - y) + 3y = 48.25 \] Simplify this to find \( y \). \[ 58.75 - 5y + 3y = 48.25 \] \[ 58.75 - 2y = 48.25 \] Subtract 48.25 from both sides: \[ 10.5 = 2y \] Divide by 2: \[ y = 5.25 \]

Step 6: Find the Value of x

Now that we have \( y = 5.25 \), substitute back into the simplified equation: \[ x + 5.25 = 11.75 \] Subtract 5.25 from both sides: \[ x = 6.5 \]

Step 7: Conclusion

The points per play for the race car simulator is \( 6.5 \), and for the snowboard simulator is \( 5.25 \).

Key Concepts

Simultaneous Equations

When tackling problems involving two or more unknowns, simultaneous equations come to the rescue. In the exercise, Marcus and Cody each provide us with different equations based on their usage of game points. Simultaneous equations involve two or more equations that share the same set of variables, and the solution is the values that satisfy all equations at once.

For instance, in this scenario, we have two equations:

• Marcus: \( 4x + 4y = 47 \)
• Cody: \( 5x + 3y = 48.25 \)

These equations help us find the values for \( x \) and \( y \) that satisfy both conditions, giving us the cost per play for each game simulator. Understanding how to identify and set up simultaneous equations is key in breaking down complex multi-variable problems.

Substitution Method

The substitution method is a powerful and straightforward technique to solve simultaneous equations. Its main idea is to isolate one variable in one equation and substitute it into another equation.

In our problem, the simplified equation from Marcus's usage, \( x + y = 11.75 \), allows us to express \( x \) in terms of \( y \):

• \( x = 11.75 - y \)

By substituting this expression for \( x \) into Cody’s equation \( 5x + 3y = 48.25 \), we essentially reduce the problem to a single equation with one variable.

This method simplifies complex systems, making it easier to find specific solutions, like determining how many points each game charges per play. Once you get the hang of substitution, it becomes an invaluable tool for problem-solving.

Problem Solving

Effective problem solving is about breaking down a problem into manageable parts. In exercises involving linear equations, this often means clearly defining variables, setting up equations, and precisely following mathematical steps to arrive at the solution.

In this example:

• Define variables: Let \( x \) be the cost for the race car and \( y \) for the snowboard.
• Create equations based on the scenarios of Marcus and Cody: \( 4x + 4y = 47 \) and \( 5x + 3y = 48.25 \).
• Use the substitution method to isolate variables and solve the equations.
• Conclude with the specific values: \( x = 6.5 \) and \( y = 5.25 \).

Each step involves logical thinking, ensuring no details are missed. By proceeding step-by-step, from defining to solving, students can tackle seemingly complicated problems with confidence and accuracy.