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. Could you find how much the carpenter is spending on 1 sheet of oak paneling? Explain.

Short answer

1 sheet of oak paneling costs 24.905 dollars.

Step-by-step solution

Setup equations

First, setup two equations for the problem using the information given: \n Equation 1 represents the cost of the first job: \(4n_1 + 2n_2 = 99.62\), where \(n_1\) stands for the price of 1 sheet of oak paneling and \(n_2\) for the price of 1 shower tileboard. \n Equation 2 represents the cost of the second job: \(12n_1 + 6n_2 = 298.86\)

Simplify equations

Now, derive Equation 3 by multiplying Equation 1 by 3, we should obtain the same values as in Equation 2 if the prices for the sheets are constant: \(12n_1 + 6n_2 = 298.86\). Comparing Equation 2 and Equation 3, we can conclude these two are identical, which indicates the tileboard does not affect the cost.

Calculate the cost of oak paneling

Now we can calculate our needed variable \(n_1\) as follows: Divide Equation 1 by 4: \(n_1 + 0.5n_2 = 24.905\). As we've previously concluded \(n_2 = 0\) we can infer \(n_1 = 24.905\).

Key concepts

Linear Equations

Linear equations are fundamental in algebra and represent one of the simplest forms of algebraic equations. They have the general form of \( ax + by = c \), where \( a \), \( b \), and \( c \), are constants, and \( x \) and \( y \) are variables representing unknown values we aim to find. In the scenario of the carpenter, the linear equations are used to represent the total cost for sheets of oak paneling and shower tileboard. Each equation corresponds to a different purchase scenario, establishing a relationship between the amounts of each item purchased and the total cost.

When dealing with linear equations, especially in a system, our goal is often to find the value of the variables in order to understand the relationship between various quantities in real-world situations, such as the pricing of materials in our carpenter's example.

Variable Isolation

Variable isolation is a technique in algebra used to solve for a single variable by rearranging the equation. To isolate a variable means to get the variable on one side of the equation and all other terms on the other side. This usually involves addition, subtraction, multiplication, or division operations applied to both sides of the equation. In our carpenter's problem, we want to isolate \( n_1 \) to find out the cost of oak paneling. By manipulating the equations, we simplify them to a point where the variable \( n_1 \) stands alone and its value can be easily determined.

Isolating variables is a step-by-step process that requires careful application of algebra rules, ensuring that whatever operation is done to one side of the equation is also done to the other, thus maintaining the equality.

Solving Algebraic Equations

Solving algebraic equations involves finding the values of the variables that make the equation true. For systems involving two equations, like in our carpenter’s case, there are several methods for solving, such as substitution or elimination. The goal is to manipulate the equations to deduce the value of one variable and then use that information to find the values of others. In the exercise provided, we used a combination of comparison and substitution. First, equations were simplified, then compared to show that \( n_2 \) had no effect on the cost, allowing us to focus on solving for \( n_1 \).

This is a crucial skill in mathematics, as it not only applies to theoretical problems but also to real-world applications like budgeting, engineering, and science, providing concrete answers from abstract representations.

Mathematical Reasoning

Mathematical reasoning involves using logic and mathematical concepts to solve problems. It's the thought process that underlies mathematics and allows us to make sense of numbers and symbols to reach conclusions. In the problem with the carpenter, mathematical reasoning was used to deduce that since multiplying the first equation by three resulted in the second equation, the second equation did not provide new information about the cost of tileboard. This indicated that the tileboard’s price is zero or not a factor, which is an insight that simplifies solving the equations.

Effective mathematical reasoning often requires identifying patterns, making assumptions, testing hypotheses, and logically justifying each step to ensure the soundness of the solution. This type of reasoning is valuable in all fields that involve quantitative analysis and critical thinking.