Problem 61 - Algebra A Combined Function

Question

The production supervisor at Alexandra's Office Products finds that it takes 3 hours to manufacture a particular office chair and 6 hours to manufacture an office desk. A total of 1200 hours is available to produce office chairs and desks of this style. The linear equation that models this situation is \(3 x+6 y=1200,\) where \(x\) represents the number of chairs produced and y the number of desks manufactured. Complete the ordered pair solution \((0, \quad)\) of this equation. Describe the manufacturing situation that corresponds to this solution.

Step-by-Step Solution

Answer

The ordered pair is (0, 200), representing 0 chairs and 200 desks manufactured.

Step 1: Substitute in the Linear Equation

We start with the linear equation \(3x + 6y = 1200\). To find the ordered pair \((0, \quad)\), we must substitute \(x = 0\) in the equation. This gives us \(3(0) + 6y = 1200\), simplifying to \(6y = 1200\).

Step 2: Solve for y

Now, solve the equation \(6y = 1200\) for \(y\) by dividing both sides by 6: \[ y = \frac{1200}{6} \]This simplifies to \(y = 200\).

Step 3: Complete the Ordered Pair

We have found that when \(x = 0\), \(y = 200\). Thus, the ordered pair is \((0, 200)\).

Step 4: Interpret the Solution

The ordered pair \((0, 200)\) signifies that the production supervisor could choose to manufacture 0 chairs and 200 desks. All available 1200 hours would be used to produce only desks, at 6 hours per desk.

Key Concepts

Ordered Pairs

When working with linear equations, you'll often come across solutions represented as ordered pairs. An ordered pair is simply a set of two numbers within parentheses, typically noted as \(x, y\). In the context of a linear equation like our example, each number in the pair corresponds to a specific variable in the equation.
For the equation \(3x + 6y = 1200\), the ordered pair \((0, 200)\) is a solution. This means when \(x = 0\), \(y\) equals 200. This is not just a random pair; it's a solution that satisfies the equation. Ordered pairs are crucial because they tell us which combination of values for \(x\) and \(y\) makes the equation true. In production planning, like our office chairs and desks scenario, these pairs can represent feasible production quantities that meet resource constraints.
Ordered pairs provide a practical way of expressing solutions visually on a graph as well. Each pair corresponds to a specific point on the plane, where the line formed by the equation crosses through. This helps to understand the relationship between variables and how they affect one another.

Substitution Method

The substitution method is a powerful tool for solving systems of equations. In a typical problem, like the production issue at Alexandra's Office Products, substitution can simplify finding solutions. Here's a simple breakdown:
First, you start with the given equation, \(3x + 6y = 1200\). You want to find the value of one variable when the other is known. For the ordered pair \(0, \_\), you substitute \(x = 0\) into the equation. This step focuses on simplifying the problem to a single-variable equation.

• Replace all instances of the chosen variable (in this case, \(x\)) with its given value.
• Re-solve the equation for the remaining variable (here, \(y\)).
• Once the value for \(y\) is clear, you have completed the pair like \(0, 200\).

This technique is quite intuitive once you get the hang of it. By reducing the equation to a single variable, it becomes much simpler. The substitution method is especially handy in production planning when you often need to decide on the amounts of different products under limited resources.

Production Planning

Production planning involves strategizing how to best use available resources to meet production goals. In the context of Alexandra's Office Products, we see this with the problems surrounding chair vs. desk production. The linear equation \(3x + 6y = 1200\) models how production is limited by time: 3 hours per chair and 6 hours per desk.
This equation serves to represent a constraint, which is an essential feature of production planning. The art of production planning involves balancing these constraints to maximize efficiency and meet output needs.
When we look at the ordered pair \(0, 200\), it tells a story about resource allocation. It means using all 1200 available hours solely for desk production. Such decisions are critical in real-world planning:

• This choice might maximize desk production if they are in higher demand or bring about higher profit margins.
• Alternatively, making no chairs could mean a shortage, impacting availability.

Understanding how linear equations and solutions like ordered pairs assist in production planning helps managers decide on production quantity based on available resources. It's about finding that sweet spot where all production goals are met efficiently and effectively.