Problem 3 - Elementary Algebra Within Reach

Question

A bakery with two stores buys three large delivery trucks and six small delivery trucks. One store receives one large delivery truck and four small delivery trucks for a total cost of \(\$ 118,000\). The second store receives two large delivery trucks and two small delivery trucks for a total cost of \(\$ 107,000\). What is the cost of each type of delivery truck?

Step-by-Step Solution

Answer

The cost of each large delivery truck is \$32000, while each small delivery truck costs \$21500.

Step 1: Format the Equations

The problem can be translated into a system of two equations. Let L represent the cost of a large truck and S represent the cost of a small truck. Based on the information, the first store equation can be written as: L + 4S = 118000 and the second store equation can be written as: 2L + 2S = 107000. To simplify equations, the second equation can be divided by 2 to get: L + S = 53500.

Step 2: Execute the Elimination

Now, we subtract the second rescaled equation from the first equation to eliminate L. This results in: L + 4S - (L + S) = 118000 - 53500, simplifying we get: 3S = 64500.

Step 3: Solve for S

We solve for S by dividing by 3: S = 64500 / 3. Therefore, S, which represents the cost of a small truck, equals \$21500.

Step 4: Solve for L

We substitute S = 21500 into the second simplified equation (L + S = 53500) to solve for L: L = 53500 - 21500. Therefore, L, representing the cost of a large truck, equals \$32000.

Key Concepts

Linear Equations

Linear equations are equations where the variables appear only to the first power and their graphs result in straight lines. In the context of our bakery problem, we're dealing with the cost represented by linear equations. Each equation represents a relationship between the costs of the trucks provided to both stores.
In the exercise, we use two simple equations based on the deliveries:

• Store 1: One large truck and four small trucks.
• Store 2: Two large trucks and two small trucks.

Each equation's structure follows the pattern of \(ax + by = c\), where \(L\) and \(S\) represent the costs of large and small trucks respectively. By rearranging the problem information in this mathematical way, we can methodically determine unknown pricing.

Substitution Method

The substitution method is a strategy used to solve systems of equations where the value of one variable is substituted from one equation into another.
In many linear equation problems, this technique helps in simplifying calculations. Here’s a quick guide for how it's done:

• Solve one of the equations for one variable.
• Substitute the expression obtained in the other equation to find the second variable.

In our bakery truck example, once we determined \(S\), the cost of a small truck was \(\$21,500\), substituting into the simplified equation \(L + S = 53,500\) helps us easily find the cost \(L\) of a large truck. This method emphasizes the power of logical replacement to find unknowns.

Elimination Method

The elimination method involves adding or subtracting equations to eliminate a variable, simplifying the process to find solutions. In this method, equations are aligned so that adding or subtracting them eliminates one variable.
For instance, in the bakery problem, once equations were prepared:

• The original equations were manipulated to yield a simpler system.
• We subtracted the revised second equation from the first to eliminate \(L\), leading to an equation in terms of \(S\) only.

With elimination, the complexity reduces, bringing us to a straightforward equation, \(3S = 64,500\). After finding \(S\), we back-trace to solve for \(L\). This method is especially useful when substitution is cumbersome.

Word Problems in Algebra

Word problems transform real-world scenarios into mathematical representations. They often involve translating the given situation into math equations for solution.
The bakery delivery truck problem is a word problem that demands comprehension of the context and relationships of costs and deliveries.

• Identify the quantities, letting \(L\) and \(S\) be your variables for the unknown costs.
• Formulate equations based on descriptive conditions presented.
• Utilize methods like substitution or elimination to solve.

Practicing word problems improves mathematical modeling skills, essential for understanding how equations are applied to describe and solve practical issues.