Sal's Sandwich Shop sells wraps and sandwiches as part of its lunch specials. The profit on every sandwich is $2, and the profit on every wrap is $3. Sal made a profit of $1,470 from lunch specials last month. The equation 2x + 3y = 1,470 represents Sal's profits last month, where x is the number of sandwich lunch specials sold and y is the number of wrap lunch specials sold.
1. Change the equation to slope-intercept form. Identify the slope and y-intercept of the equation. Be sure to show all your work.
2. Describe how you would graph this line using the slope-intercept method. Be sure to write using complete sentences.
3. Write the equation in function notation. Explain what the graph of the function represents. Be sure to use complete sentences.
4. Graph the function. On the graph, make sure to label the intercepts. You may graph your equation by hand on a piece of paper and scan your work or you may use graphing technology.
5. Suppose Sal's total profit on lunch specials for the next month is $1,593. The profit amounts are the same: $2 for each sandwich and $3 for each wrap. In a paragraph of at least three complete sentences, explain how the graphs of the functions for the two months are similar and how they are different.
1. To change the equation to slope-intercept form (y = mx + b), we need to isolate y.
Start with the equation: 2x + 3y = 1,470.
First, subtract 2x from both sides: 3y = -2x + 1,470.
Next, divide both sides by 3 to isolate y: y = (-2/3)x + 490.
The slope (m) of the equation is -2/3, and the y-intercept (b) is 490.
2. To graph the line using the slope-intercept method, follow these steps:
- Start by plotting the y-intercept, which is the point (0, 490).
- Then, use the slope to find additional points. Since the slope is -2/3, you can move "down 2 units and to the right 3 units" to find another point.
- Connect the points with a straight line to complete the graph.
3. The equation in function notation is f(x) = (-2/3)x + 490. This represents a linear function where the input is the number of sandwich lunch specials (x), and the output f(x) represents the profit made from selling those sandwiches. The coefficient of x (-2/3) represents the rate of change or the slope of the function. The constant term (490) represents the starting profit when no sandwich lunch specials are sold.
4. Unfortunately, as an AI text-based bot, I am unable to directly graph the function for you. However, you can graph it by following the steps outlined in question 2. Remember to label the intercepts, which are (0, 490) for the y-intercept.
5. If Sal's profit for the next month is $1,593, we can use the same equation, f(x) = (-2/3)x + 490, to find the number of sandwich lunch specials (x) needed to achieve that profit.
Rearranging the equation to solve for x:
(-2/3)x + 490 = 1,593
Subtract 490 from both sides:
(-2/3)x = 1,103
Multiply both sides by -3/2 to isolate x:
x = -1,103 * (-3/2)
x ≈ 1,654
So, Sal would need to sell approximately 1,654 sandwich lunch specials in order to reach a profit of $1,593.
Comparing the two graphs of the function for the two months, we can observe that they have the same slope (-2/3), indicating the same rate of change in profit. However, the y-intercepts are different due to the different starting profits. The graphs are linear and resemble straight lines, but they are shifted vertically on the y-axis. This difference in y-intercepts reflects the varying starting profits.