You have $47 to spend on music and movie downloads. Each album download costs $7 and each movie download costs $8. Write and graph a linear inequality that represents this situation. Let x represent the number of albums and y the number of movies.
each album costs $7 and you buy x of them
cost of albums = 7x
each movie costs $8 and you have you have y of them
cost of movies = 8y
total cost is has to be less than $47
---> 7x + 8y < 47
sketch the graph of 7x + 8y = 47 , a straight line, and shade in the region below that line. Of course you would stay in the first quadrant, (x > 0, and y >0 , since you can't have negative number of movies and albums)
Since you can't buy partial albums and movies, both x and y have to be whole numbers.
So for solutions you would consider only ordered pairs that have whole numbers as their components.
e.g. (2, 2) or (1,5)
Well, let's start with the cost of the albums and movies. Since each album download costs $7 and each movie download costs $8, we can write the total cost as:
Cost = 7x + 8y
Next, let's consider the budget constraint. We have $47 to spend, so the total cost cannot exceed this amount. Therefore, we can write the inequality as:
7x + 8y ≤ 47
Now, let's graph this inequality on a coordinate plane:
Y
|
|
|
- --------(0,5)-----------(5.5,0)----------
In this graph, the x-axis represents the number of albums (x) and the y-axis represents the number of movies (y). The line represents the equation 7x + 8y = 47. The area below or on the line satisfies the inequality.
Note that the points along the line are also valid solutions, as they would equal exactly $47. However, since the task involves spending the entire budget, it is more realistic to choose a point slightly below the line.
Let's start by writing the inequality.
The cost of x album downloads is 7x, and the cost of y movie downloads is 8y. Since we have $47 to spend, the total cost of the downloads must be less than or equal to $47. Therefore, the inequality is:
7x + 8y ≤ 47
To graph this inequality, we will need to plot the boundary line and shade the region that satisfies the inequality.
To plot the boundary line, we can rewrite the inequality in slope-intercept form:
8y ≤ -7x + 47
y ≤ -7/8x + 47/8
To graph the line, we can start by plotting two points on the line. Let's set x = 0:
y = -7/8(0) + 47/8 = 47/8
So the first point is (0, 47/8).
And let's set y = 0:
0 = -7/8x + 47/8
7/8x = 47/8
x = 47/7
So the second point is (47/7, 0).
Now we can draw a line passing through these two points.
After drawing the line, we need to shade the region that satisfies the inequality. Since the inequality is "less than or equal to," we need to shade below the line.
Therefore, the graph of the inequality 7x + 8y ≤ 47 is a line with a negative slope passing through (0, 47/8) and (47/7, 0), and the region below the line is shaded.
To write a linear inequality that represents the given situation, we need to consider two constraints:
1. The total cost of the album downloads cannot exceed $47.
2. The total cost of the movie downloads cannot exceed $47.
Let's break down each constraint:
1. The cost of x album downloads: 7x
- Since each album download costs $7, the total cost of x album downloads is 7x.
2. The cost of y movie downloads: 8y
- Since each movie download costs $8, the total cost of y movie downloads is 8y.
Now, let's write the linear inequality for each constraint:
1. The total cost of the album downloads (7x) should not exceed $47:
- 7x ≤ 47
2. The total cost of the movie downloads (8y) should not exceed $47:
- 8y ≤ 47
Combining both inequalities, we can write the linear inequality in terms of x and y as:
- 7x + 8y ≤ 47
To graph this inequality, we can convert it into slope-intercept form:
- 8y ≤ -7x + 47
Dividing all terms by 8 (to isolate y), we have:
- y ≤ (-7/8)x + 47/8
Now we can graph this inequality on a coordinate plane.