Problem 64 - 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. If 50 chairs are manufactured, find the greatest number of desks that can be made.

Step-by-Step Solution

Answer

175 desks can be made if 50 chairs are produced.

Step 1: Understand the Problem

We need to find the number of desks that can be manufactured if 50 chairs are produced, given the equation \(3x + 6y = 1200\). Here, \(x\) represents the number of office chairs, and \(y\) represents the number of office desks.

Step 2: Substitute Known Value

We know the number of chairs \(x = 50\). Substitute this value into the equation: \(3(50) + 6y = 1200\).

Step 3: Simplify the Equation

Calculate \(3 \times 50 = 150\). Substitute back to get: \(150 + 6y = 1200\).

Step 4: Isolate Variable y

Subtract 150 from both sides to isolate the \(6y\) term: \(6y = 1200 - 150\).

Step 5: Solve for y

Calculate \(1200 - 150 = 1050\), so the equation becomes \(6y = 1050\). Divide both sides by 6 to solve for \(y\): \(y = \frac{1050}{6}\).

Step 6: Calculate the Solution

Perform the division: \(y = 175\). So, the greatest number of desks that can be made is 175.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a set of unknowns, which we aim to solve simultaneously. In our example, we have a single equation, but it models a straightforward system scenario involving two types of production: chairs and desks. When dealing with systems in linear equations, the goal is to find values for the unknowns that satisfy all the given equations. This can be visualized as finding a point where two lines intersect, but more terms might solve multiple equations. Understanding systems of equations is crucial because they allow us to model real-world problems where multiple conditions must be met at once. By using these systems, we can explore relationships between different variables, such as time, resources, and products.

Solving Equations

Solving equations involves finding the value of the variable that makes the equation true. For linear equations like the one in our exercise, the process includes substituting, simplifying, and isolating variables.

• **Substitution**: We insert known values into the equation to simplify it. In our case, we substituted the number of chairs produced (50) into the equation.
• **Simplification**: We break complex equations into simpler forms. Here, multiplying and subtracting allowed us to isolate the desk variable.
• **Isolation and Solution**: By isolating the variable we are interested in, we find its value. Dividing the equation at this step helps us reach the final answer for the number of desks produced: 175.

Each of these steps is important because they help manage complex scenarios involving multiple variables, simplifying them into more understandable parts.

Real-World Applications

Linear equations are not just theoretical; they have numerous practical applications in the real world, including business, economics, engineering, and more. Our problem is a clear example of production planning in a manufacturing context where resources are finite. In the example, linear equations help allocate production resources optimally, ensuring time is maximized without overextending limits. This type of problem is common where manufacturing timelines must align with operational capacity. Using linear equations in such contexts helps make predictions, optimize operations, and even forecast financial outcomes. This is instrumental in decision-making processes where constraints and objectives must be balanced effectively.