Problem 63 - 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 desks are manufactured, find the greatest number of chairs that can be made.

Step-by-Step Solution

Answer

300 chairs can be made.

Step 1: Set up the equation

We begin with the equation given in the problem: \(3x + 6y = 1200\). This equation expresses the total production hours available for making chairs and desks, where \(x\) is the number of chairs and \(y\) is the number of desks.

Step 2: Substitute the number of desks

We know that 50 desks are manufactured, which means \(y = 50\). Substitute this value into the equation: \(3x + 6(50) = 1200\).

Step 3: Simplify the equation

Calculate the total hours used for manufacturing 50 desks: \(6 \times 50 = 300\). Substitute this into the equation: \(3x + 300 = 1200\).

Step 4: Solve for x

Subtract 300 from both sides of the equation to isolate the term involving \(x\): \(3x = 1200 - 300\), which simplifies to \(3x = 900\).

Step 5: Divide to find the number of chairs

To solve for \(x\), divide both sides of the equation by 3: \(x = \frac{900}{3}\). This simplifies to \(x = 300\). Thus, the greatest number of office chairs that can be made is 300.

Key Concepts

Algebra and Linear Equations

Algebra is a branch of mathematics focused on using symbols and letters to represent numbers in equations and formulas. In our exercise about Alexandra's Office Products, we're dealing with a linear equation. A linear equation is an equation with variables that have a maximum power of one. In simple terms, it forms a straight line when graphed on a coordinate plane. In the equation given, \(3x + 6y = 1200\), we're dealing with two variables: \(x\) and \(y\). These represent how many office chairs and desks can be produced respectively. The coefficients 3 and 6 indicate how many hours are needed per item type, and the total 1200 represents the maximum hours available. This setup is common in algebra when representing real-life situations like production planning.

Problem-Solving Strategy

Problem-solving is the process of finding solutions to difficult or complex issues. In our exercise, we're tasked with finding how many chairs can be made if a certain number of desks are already being produced. To solve this problem, follow these steps:

• Start by understanding the problem. Know what's being asked and what information is given.
• Translate the word problem into a mathematical equation or representation.
• Use relevant mathematical techniques, such as substitution, to simplify the problem.
• Solve the equation step by step until you find the desired result. In this case, the maximum number of chairs.

This approach helps break the problem into manageable parts and makes complex problems simpler.

Substitution Method

The substitution method is a widely used technique in algebra to solve systems of equations. It involves solving a system by replacing one variable with an expression derived from another equation. For problems where one of the variables has been specified, like the number of desks in our exercise, this method comes in handy.Here's how we applied the substitution method:

• We know from the exercise that 50 desks, represented by \(y\), are produced.
• We substitute \(y = 50\) into the equation \(3x + 6y = 1200\).
• This leads to \(3x + 6(50) = 1200\).
• The equation now only contains one variable, making it easier to solve for \(x\).

This method simplifies finding the solution because it reduces the number of variables in the equation.

Equation Simplification

Simplifying an equation is a key step in solving it effectively. Simplification involves making the equation as straightforward as possible by combining like terms or eliminating unnecessary components.In the exercise, after substituting the desks into the equation, we have \(3x + 300 = 1200\). Here's how simplification aids in solving:

• First, subtract 300 from both sides to eliminate extra terms: \(3x = 900\).
• This creates a much simpler one-step equation with only the variable \(x\) to solve for.
• Finally, divide both sides by 3 to isolate \(x\): \(x = \frac{900}{3} = 300\).

Simpler equations are easier to solve, reducing the chance for errors and making solutions more intuitive.