Answer :
To determine which equation models the total profit, [tex]\( y \)[/tex], based on the number of magazines sold, [tex]\( x \)[/tex], we start by analyzing the information given:
1. After the journalism club sold 60 magazines, it had [tex]$220 in profit.
2. After it sold a total of 100 magazines, it had a total of $[/tex]420 in profit.
Based on this information, we recognize that the relationship between the number of magazines sold and the profit is linear. We can describe this relationship with a linear equation of the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( y \)[/tex] is the total profit,
- [tex]\( x \)[/tex] is the number of magazines sold,
- [tex]\( m \)[/tex] is the slope of the line (profit per magazine),
- [tex]\( b \)[/tex] is the y-intercept (the initial profit when no magazines are sold).
### Step 1: Calculate the Slope
The slope [tex]\( m \)[/tex] is the change in profit per magazine sold. It can be calculated using the given points:
- Point 1: (60 magazines, [tex]$220 profit)
- Point 2: (100 magazines, $[/tex]420 profit)
The slope formula is:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{420 - 220}{100 - 60} = \frac{200}{40} = 5 \][/tex]
Thus, the slope [tex]\( m = 5 \)[/tex]. This means each magazine sold adds $5 to the profit.
### Step 2: Create the Equation
We use the point-slope form of the equation to create our line. The point-slope formula is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Let's use the first point (60, 220):
[tex]\[ y - 220 = 5(x - 60) \][/tex]
This equation fits the form given in the options.
### Step 3: Choose the Correct Equation
Rewriting the equation:
[tex]\[ y - 220 = 5(x - 60) \][/tex]
Compare this to the provided options:
- A. [tex]\( y + 220 = 2(x + 60) \)[/tex]
- B. [tex]\( y + 220 = 5(x + 60) \)[/tex]
- C. [tex]\( y - 220 = 5(x - 60) \)[/tex]
- D. [tex]\( y - 220 = 2(x - 60) \)[/tex]
The equation we derived, [tex]\( y - 220 = 5(x - 60) \)[/tex], matches option C.
Therefore, the correct equation is:
C. [tex]\( y - 220 = 5(x - 60) \)[/tex]
1. After the journalism club sold 60 magazines, it had [tex]$220 in profit.
2. After it sold a total of 100 magazines, it had a total of $[/tex]420 in profit.
Based on this information, we recognize that the relationship between the number of magazines sold and the profit is linear. We can describe this relationship with a linear equation of the form:
[tex]\[ y = mx + b \][/tex]
where:
- [tex]\( y \)[/tex] is the total profit,
- [tex]\( x \)[/tex] is the number of magazines sold,
- [tex]\( m \)[/tex] is the slope of the line (profit per magazine),
- [tex]\( b \)[/tex] is the y-intercept (the initial profit when no magazines are sold).
### Step 1: Calculate the Slope
The slope [tex]\( m \)[/tex] is the change in profit per magazine sold. It can be calculated using the given points:
- Point 1: (60 magazines, [tex]$220 profit)
- Point 2: (100 magazines, $[/tex]420 profit)
The slope formula is:
[tex]\[ m = \frac{\Delta y}{\Delta x} = \frac{420 - 220}{100 - 60} = \frac{200}{40} = 5 \][/tex]
Thus, the slope [tex]\( m = 5 \)[/tex]. This means each magazine sold adds $5 to the profit.
### Step 2: Create the Equation
We use the point-slope form of the equation to create our line. The point-slope formula is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Let's use the first point (60, 220):
[tex]\[ y - 220 = 5(x - 60) \][/tex]
This equation fits the form given in the options.
### Step 3: Choose the Correct Equation
Rewriting the equation:
[tex]\[ y - 220 = 5(x - 60) \][/tex]
Compare this to the provided options:
- A. [tex]\( y + 220 = 2(x + 60) \)[/tex]
- B. [tex]\( y + 220 = 5(x + 60) \)[/tex]
- C. [tex]\( y - 220 = 5(x - 60) \)[/tex]
- D. [tex]\( y - 220 = 2(x - 60) \)[/tex]
The equation we derived, [tex]\( y - 220 = 5(x - 60) \)[/tex], matches option C.
Therefore, the correct equation is:
C. [tex]\( y - 220 = 5(x - 60) \)[/tex]