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.
Change the equation to slope-intercept form. Identify the slope and y-intercept of the equation. Be sure to show all your work.
Describe how you would graph this line using the slope-intercept method. Be sure to write using complete sentences.
Write the equation in function notation. Explain what the graph of the function represents. Be sure to use complete sentences.
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.
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.
Below is a graph that represents the total profits for a third month. Write the equation of the line that represents this graph. Show your work or explain how you determined the equations.
2x + 3y = 1470
Slope intercept form is y = mx+b
So you want to solve for y:
subtract 2x from both sides and divide by 3.
3y = -2x +1470
y = -2/3 x + 1470/3 or y = -2/3x +490
To graph you would start at the point (0,490) and to find the next point you would use -2/3 = change in y/change in x
so you could move -2 on the y and +3 on the x to find you next point. Repeat this again to find your 3rd point.
what about numbers 5 and 6?
To change the equation 2x + 3y = 1,470 to slope-intercept form, we need to isolate y on one side of the equation.
Starting with the equation:
2x + 3y = 1,470
Step 1: Subtract 2x from both sides to isolate 3y:
3y = 1,470 - 2x
Step 2: Divide both sides by 3 to solve for y:
y = (1,470 - 2x) / 3
Now that we have the equation in slope-intercept form (y = mx + b), we can identify the slope and y-intercept.
The slope is the coefficient of x, which is -2/3.
The y-intercept is the constant term, which is 1,470/3.
To graph the line using the slope-intercept method, we need to plot the y-intercept (where the line crosses the y-axis) and use the slope to find other points.
The y-intercept is (0, 490) since the equation gives us y = 1,470/3 when x = 0.
To find another point, we can use the slope. The slope of -2/3 means that for every 3 units we move to the right along the x-axis, we move down 2 units.
So, starting from the y-intercept (0, 490), we can move 3 units to the right (0 + 3 = 3) and then move down 2 units (490 - 2 = 488). This gives us the point (3, 488).
We can continue this process to find more points and then plot them on a graph.
The equation in function notation would be:
f(x) = (1,470 - 2x) / 3
The graph of this function represents the profit (in dollars) from selling lunch specials based on the number of sandwich specials sold (x) and wrap specials sold (y). The x-axis represents the number of sandwich specials, the y-axis represents the number of wrap specials, and the points on the graph represent the profit earned for different combinations of sandwich and wrap specials.
To graph the function, you can use graphing technology or follow the steps mentioned earlier to plot points and connect them to form a line. Make sure to label the intercepts (y-intercept and x-intercept) on the graph.
Given the profit of $1,593 for the next month with the same profit amounts per sandwich and wrap ($2 and $3), the slope-intercept equation will remain the same: y = (1,470 - 2x) / 3. However, the values for x and y will be different. The graphs of the functions for the two months will be similar in shape and slope but will have different intercepts due to the difference in profit amounts.
Unfortunately, you haven't provided the graph that represents the total profits for a third month. Without the graph, it is not possible to write the equation of the line that represents it. However, you can determine the equation of the line by finding the slope and y-intercept from the given graph.