- Construct the word problem by identifying 3 items at price \(x\) and a fixed cost of \(\$15\) totaling \(\$84\).
- Solve the equation \(3x + 15 = 84\) by subtracting 15 from both sides to get \(3x = 69\).
- Divide by 3 to find \(x = \boxed{23}\).
- Interpret \(x\) as the unit price of the items purchased.
Explanation
- Analysis of the Problem
The given equation is \(3x + 15 = 84\). This is a linear equation in one variable \(x\). The equation represents a situation where a base cost of 15 is added to three times a value \(x\), resulting in a total of 84. To solve this, we will isolate \(x\) by performing operations on both sides of the equation. First, we subtract 15 from both sides to account for the fixed cost. Then, we divide the resulting amount by 3 to determine the value of the individual unit represented by \(x\).
- Part A: Word Problem Construction
Suppose you are shopping at a bookstore and you decide to buy 3 identical journals. In addition to the cost of the journals, you are charged a flat shipping fee of \(\$15\). If the total amount you paid for the order is \(\$84\), we can write the following equation to represent this transaction:
\[3x + 15 = 84\]
In this context, \(x\) represents the price of each individual journal in dollars.
- Part B: Solving the Equation
To solve the equation \(3x + 15 = 84\):
- Subtract 15 from both sides of the equation:
\[3x + 15 - 15 = 84 - 15\]
\[3x = 69\]
- Divide both sides by 3 to isolate \(x\):
\[x = \frac{69}{3}\]
\[x = 23\]
Thus, the value of the variable is \(x = 23\).
- Part C: Explanation of the Variable
The variable \(x\) represents the cost of a single journal. Based on the solution found in Part B, each journal costs \(\$23\). This aligns with the total cost: buying 3 journals at \(\$23\) each costs \(\$69\), and adding the \(\$15\) shipping fee brings the total to \(\$84\).
Examples
Linear equations like \(3x + 15 = 84\) are fundamental in everyday financial planning. For instance, if you are planning a birthday party where there is a flat venue rental fee (\(15\)) and a per-person cost (\(3x\)) for catering, you can use an equation to determine how many people you can invite if you have a set budget of \(\$84\). Mastering this skill allows you to manage resources and expenses effectively in daily life.