Problem 38 - Algebra and Trigonometry

Question

Use the four-step strategy to solve each problem. Use \(x, y,\) and \(z\) to represent unknown quantities. Then translate from the verbal conditions of the problem to a system of three equations in three variables. A certain brand of razor blades comes in packages of \(6,12\) and 24 blades, costing \(\$ 2, \$ 3,\) and \(\$ 4\) per package, respectively. A store sold 12 packages containing a total of 162 razor blades and took in \(\$ 35 .\) How many packages of each type were sold?

Step-by-Step Solution

Answer

The solution to this system of equations will provide the exact number of each type of package sold which is what the original problem is asking for.

Step 1: Represent the unknowns

Let \(x\) represent the number of 6-blade packages, \(y\) represent the number of 12-blade packages, and \(z\) represent the number of 24-blade packages.

Step 2: Translate the problem's conditions into equations

Firstly, the total number of packages sold is 12, thus the equation is \(x + y + z = 12\). Secondly, the total number of blades sold is 162. Each type of package contains a different number of blades, so the equation becomes \(6x + 12y + 24z = 162\). Lastly, the total revenue earned is $35, so, given the different costs for the different packages, the final equation is \(2x + 3y + 4z = 35\).

Step 3: Solve the system of equations

This system of equations can be solved via substitution, elimination, or matrix methods.

Step 4: Continue solving the system of equations

Iteratively continue solving the system of equations. Due to the multiple steps required within this step, it may be beneficial to subdivide it into smaller sub-steps.

Step 5: Find the solution

The solution to this system of equations will provide the respective values of \(x\), \(y\) and \(z\). These values represent the number of packages of each type that were sold.

Key Concepts

Linear Equations

Linear equations are mathematical statements that involve variables raised only to the first power with constant coefficients. They take a general form of: \[ ax + by + cz = d \] where \(a\), \(b\), and \(c\) are coefficients and \(x\), \(y\), \(z\) are the variables. In our razor blade problem, each equation is linear because none of the variables are squared or involve higher powers. A linear equation usually has one or more terms set equal to a constant.

• Simple to manipulate.
• Follow basic arithmetic operations: addition, subtraction, multiplication, and division.
• Used to form systems when working with multiple equations.

Linear equations are excellent for representing relationships between different quantities, and they can be solved analytically using various algebraic methods.

Algebraic Methods

Algebraic methods are techniques used to find solutions to equations. When working with systems of equations, as in our example, you have several options:

• Substitution: Solve one of the equations for a variable, then substitute that expression into the other equations. Repeat this until only one variable remains.
• Elimination: Add or subtract equations to eliminate one variable at a time, reducing the system to fewer equations.
• Matrix Methods: Such as Gaussian elimination or using matrix inverses, which are particularly helpful for larger systems.

In our exercise, any of these methods can solve the system of three linear equations effectively. Algebraic methods allow you to systematically break down complex systems into simpler steps.

Unknown Variables

In algebra, unknown variables are symbols that represent quantities we aim to find. In the razor blade exercise, \(x\), \(y\), and \(z\) are the unknown variables.

• \(x\) represents the number of 6-blade packages sold.
• \(y\) represents the number of 12-blade packages sold.
• \(z\) represents the number of 24-blade packages sold.

These variables allow us to create equations from verbal conditions. Without them, translating a word problem into mathematical language would be impossible. Keeping track of what each variable represents is crucial when solving any problem involving several unknowns.

Problem-Solving Strategy

A structured problem-solving strategy involves taking a systematic approach to understand and solve a problem. In the given exercise, we follow these steps:

• Step 1: Define unknowns and choose variables.
• Step 2: Translate the problem into a system of linear equations.
• Step 3: Select an algebraic method to solve the equations.
• Step 4: Execute each step methodically to simplify equations and find the solution.
• Step 5: Interpret the solution to make sure it makes sense in the context of the problem.

Following a clear strategy not only simplifies complex problems but also ensures no steps or details are overlooked. It's a key part of getting accurate and efficient solutions.