Question 8·Medium·Linear Equations in One Variable
A school club is selling candles to raise money. Each candle sold brings in a certain amount of profit, and the club had to pay an initial cost of $150 for supplies. The equation above models how many candles must be sold to reach a total profit of $300. Which of the following is the best interpretation of the number 6 in this context?
For equation-interpretation questions, start by identifying what the variable represents and then translate each term of the equation into words. Constants usually represent fixed amounts (like start-up costs or final goals), while coefficients attached to variables represent "per unit" rates (like dollars per item or miles per hour). Use the units and the story context to decide which part of the situation each number matches, and eliminate choices that actually describe a different number or the solution rather than the coefficient itself.
Hints
Identify what the variable stands for
Look back at the problem statement: what does represent in words? Keep that in mind when you look at the term .
Match each number to the story
The problem already tells you that is the initial cost and is the profit goal. That means has to represent the remaining part of the story. What quantity would depend on how many candles are sold?
Think about units
The $150 and $300 are in dollars. Since is the number of candles, what units must the 6 have so that is also in dollars?
Desmos Guide
Graph the profit equation
In Desmos, enter the equation as y = 6x - 150. Think of as the number of candles and as the total profit in dollars.
Use a table to see how profit changes
Click on the gear icon next to the equation and select "Table" (or add a table for the equation). Look at how the -value (profit) changes when (candles) increases by 1. Notice that the profit increases by the same amount for each additional candle; match that constant increase with the correct interpretation among the answer choices.
Step-by-step Explanation
Understand what each symbol in the equation represents
The equation is .
- is the number of candles sold.
- The left side, , represents the total profit after paying for supplies.
- The right side, , is the profit goal in dollars.
- The is the initial cost for supplies in dollars (given directly in the problem).
Interpret the structure of
Think about how profit is usually calculated: it is
Here:
- is the fixed cost.
- So must represent the total profit from all candle sales before subtracting the .
- Since is the number of candles, is the amount attached to each candle and is being multiplied by the number of candles.
Use units to identify what the 6 means
The terms and are both in dollars. The term must also be in dollars. Because is in candles, the number 6 must be in dollars per candle, meaning the profit in dollars made from each candle sold.
So, the best interpretation of the number 6 is: the amount of profit, in dollars, made from each candle sold.