Question

Use the following information. A carpenter is buying supplies for a job. The carpenter needs 4 sheets of oak paneling and 2 sheets of shower tileboard. The carpenter pays 99.62 dollars for these supplies. For the next job the carpenter buys 12 sheets of oak paneling and 6 sheets of shower tileboard and pays 298.86 dollars. If the carpenter later spends a total of 139.69 dollars for 1 sheet of shower tileboard and 8 sheets of oak paneling, could you find how much 1 sheet of oak paneling costs? Explain.

Short answer

The problem seems to contain an inconsistency in the given data. Normally, one would solve the system of equations to find the cost of one sheet of oak paneling, but in this case, the calculations led to an unrealistic solution. It's highly likely that there's a mistake in the pricing of the jobs given in the problem.

Step-by-step solution

Define the Variables

Let's call the cost of one sheet of oak paneling \(x\) and the cost of one sheet of shower tileboard \(y\). Adding these costs for each of the carpenter's jobs will give us the total cost for each job. This allows us to represent the carpenter's purchases as system of linear equations.

Write Down the System of Equations

Using the above defined variables \(x\) and \(y\), the equations based on problem would be:\n\nFor the first job: \(4x + 2y = 99.62\)\n\nFor the second job: \(12x + 6y = 298.86\)

Simplify the Equations

The second equation can be simplified if it's divided by 6, to get \(2x + y = 49.81\). Now we have two simplified equations: \n\n\(4x + 2y = 99.62\)\n\n\(2x + y = 49.81\)

Solve the System of Equations

We solve it by subtraction, subtracting the second equation from the first leads to:\n\n\(2x + y = 49.81\)\n\n-subtracting from\n\n\(4x + 2y = 99.62\)\n\nWe get \(2x + y = 0\) which simplifies to \(x = 0\). This equation indicates that the cost of one sheet of oak paneling is 0, which is not true considering the actual job prices presented. Thus, there is an error in the problem.

Error Analysis

If we look at the jobs given, the second job is exactly three times the materials of the first job, thus the cost should also be three times. But the price stated for the second job is not three times the cost of the first. Therefore, one of the job prices is given incorrectly, causing the mathematical inconsistency.

Key concepts

Algebraic Equations

Algebraic equations are fundamental in algebra and serve as the bedrock for advanced mathematical reasoning. An algebraic equation is a mathematical statement that asserts the equality of two expressions, typically involving variables and constants. These equations are comprised of terms, which can be numbers, variables (representing unknown values), or the product of numbers and variables.

For instance, the equation given in the carpenter's problem, 4x + 2y = 99.62, is an algebraic equation with 'x' and 'y' as its variables. This particular form of an equation, where the highest power of the variable is one, is known as a linear equation. The primary purpose of algebraic equations in the problem-solving process is to represent real-life situations algebraically to find the values of the unknowns, such as the cost of one sheet of oak paneling or shower tileboard in the given exercise.

By creating equations from the given problem conditions, we can use algebraic methods to solve for the variables. Mistakes in the determination of these equations, like in the problem with the carpenter, can affect the entire solving process, leading us to incorrect or impossible results.

Variable Definition in Algebra

In algebra, variables are symbols, often letters, used to represent unknown or changeable values. They are placeholders for quantities that we are either trying to find or that can vary. Defining variables is an essential first step when translating word problems into algebraic expressions or equations; it clarifies what each symbol denotes and helps organize the problem-solving process.

In the context of the carpenter's problem, variables are defined as 'x' for the cost of one sheet of oak paneling and 'y' for the cost of one sheet of shower tileboard. This variable definition allows us to create formulas, algebraic equations, which we can then manipulate to find the values that satisfy all conditions of the problem.

Correctly defining variables is crucial because a misstep at this stage can lead to confusion and incorrect solutions. A well-defined variable also helps ensure clarity and precision, making it easier to communicate the problem and solution to others.

Solving Linear Systems

Solving linear systems is the process of finding the values of the variables that satisfy all equations in the system simultaneously. These systems can arise naturally from real-world problems where multiple conditions must be met at the same time, just like in the carpenter example.

There are multiple methods for solving systems of linear equations, including graphing, substitution, elimination, and matrix operations. The elimination method is partially utilized in the provided solution, aiming to simplify the system of equations through subtraction or addition to eventually isolate one of the variables.

However, a key step in solving such systems is ensuring the equations are consistent and accurate. If there is an error in the formation of the equations, as highlighted in Step 5 of the exercise solution, it's impossible to find a solution because the system would be inconsistent or have no solution at all. This is why examining the correctness of equations and the proportionality of their terms—ensuring they scale correctly with respect to the real-world quantities they represent—is an important verification step before attempting to solve them.