Problem 26 - Algebra 2

Question

Beatriz is checking a shipment of technology equipment that contains laser printers that cost \(\$ 700\) each and color monitors that cost \(\$ 200\) each. She counts 30 boxes on the loading dock. The invoice states that the order totals \(\$ 15,000\) . Write a system of two equations that represents the number of each item.

Step-by-Step Solution

Answer

The system of equations is \( x + y = 30 \) and \( 700x + 200y = 15000 \).

Step 1: Define Variables

Let's define variables for the unknown quantities. Let \( x \) represent the number of laser printers and \( y \) represent the number of color monitors.

Step 2: Set Up the First Equation

Based on the total number of boxes, we know that the sum of the laser printers and color monitors is equal to 30. Therefore, our first equation is: \[ x + y = 30 \]

Step 3: Set Up the Second Equation

Now, considering the total cost of \( \$15,000 \) and the prices of laser printers and color monitors, we can create the second equation: \[ 700x + 200y = 15000 \]

Step 4: Verify the System of Equations

Ensure the equations accurately represent the problem. The two equations \( x + y = 30 \) (count of boxes) and \( 700x + 200y = 15000 \) (total cost) correctly describe the scenario with respect to the number of items and their costs.

Key Concepts

Variables Definition

In any mathematical problem involving systems of linear equations, defining variables is an essential first step. This helps us translate a word problem into a mathematical form. In our exercise, we are dealing with two types of equipment: laser printers and color monitors. These are the unknowns that we need to determine. To simplify the process:

• Let's denote the number of laser printers by the variable \( x \).
• We will represent the number of color monitors by the variable \( y \).

This method of assigning variables not only aids in setting up equations but also makes it easier to communicate and solve the problem with clarity.

Equation Verification

Having established our equations, we must verify whether they correctly represent the given situation. Verification ensures that the problem has been accurately transformed into mathematical language, reflecting all given conditions. For our exercise, we have:

• The total count equation: \( x + y = 30 \), representing the overall number of boxes, which is clear since any combination of printers and monitors must fill these boxes.
• The cost equation: \( 700x + 200y = 15000 \), ensuring the monetary sum correlates with the given costs of the printers and monitors multiplied by their respective counts.

Both equations must hold simultaneously for our solution to be correct. This step is crucially about validation, where logical consistency with the problem context is checked.

Problem Representation

Problem representation in algebra involves translating a real-world scenario into a mathematical framework. It's crucial for understanding how to manipulate and solve the situation. In our example:- We interpret the count of items with the equation \( x + y = 30 \). - The total purchase amount is translated into the equation \( 700x + 200y = 15000 \).These equations succinctly capture all the problem's key details. The first equation handles the physical reality of the total number of boxes, while the second captures the financial requirements of the situation. Together, they create a system that models our scenario effectively. Proper representation ensures we have ready access to key insights of a problem with clarity and purpose.

Algebraic Thinking

Algebraic thinking is a cognitive process that requires one to think abstractly and critically about numbers and their relationships. When dealing with systems of linear equations, algebraic thinking involves:

• Identifying relationships between different quantities, such as prices and counts of items.
• Translating these relationships into mathematical form, i.e., creating equations that model these relationships.
• Solving these equations to find the values of unknown variables, like our \( x \) and \( y \).

This approach helps you understand not just how to solve the problem at hand, but also how similar problems can be tackled. It develops your skills in reasoning, problem-solving, and seeing the interconnectedness of numerical relationships, thus enhancing your overall mathematical competence.