Gianna is a songwriter who collects royalties on her songs whenever they are played in a commercial or a movie. Gianna will earn $50 every time one of her songs is played in a commercial and she will earn $100 every time one of her songs is played in a movie. Gianna's songs were played on twice as many commercials as movies and her total earnings on the royalties from all commercials and movies was $800. Graphically solve a system of equations in order to determine the number of commercials, x, and the number of movies, y, on which Gianna's songs were played.
Give the coordinates
(x,y) of the solution to the system of equations, where x represents the number of commercials and y represents the number of movies on which Gianna's songs were played.
The system of equations can be expressed as follows:
1) x + y = total number of songs played (equation relating the total number of songs played)
2) 50x + 100y = total earnings from royalties (equation relating the total earnings)
Since we know that Gianna's songs were played on twice as many commercials as movies, we can also express this relationship as:
3) x = 2y (equation relating the number of commercials and movies)
Solving the system of equations, we can substitute equation 3 into equations 1 and 2:
1) (2y) + y = total number of songs played
=> 3y = total number of songs played
=> y = (1/3) * (total number of songs played)
2) 50(2y) + 100y = total earnings from royalties
=> 100y + 100y = total earnings from royalties
=> 200y = total earnings from royalties
=> y = (1/200) * (total earnings from royalties)
Substituting back into equation 1, we have:
(2y) + y = (1/3) * (total number of songs played)
=> 3y = (1/3) * (total number of songs played)
=> y = (1/9) * (total number of songs played)
Therefore, the coordinates (x,y) of the solution to the system of equations are:
(x, y) = (2y,y) = (2(1/9)*(total number of songs played), (1/9)*(total number of songs played))
Since the total earnings from royalties was $800, the total number of songs played can be determined by dividing the total earnings by $50 (since each commercial earns $50):
(total number of songs played) = (total earnings from royalties) / $50
= $800 / $50
= 16
Substituting this value into the coordinates, we have:
(x, y) = (2(1/9)*16, (1/9)*16)
= (32/9, 16/9)
Are you sure this is right?
Apologies for the previous incorrect response. Let's solve the system of equations correctly:
1) x + y = total number of songs played
2) 50x + 100y = total earnings from royalties
3) x = 2y
First, substitute equation 3 into equation 1:
2y + y = total number of songs played
3y = total number of songs played
y = (1/3) * (total number of songs played)
Next, substitute y back into equation 3:
x = 2 * (1/3) * (total number of songs played)
x = (2/3) * (total number of songs played)
Now, substitute x and y back into equation 2 to solve for the total earnings from royalties:
50((2/3) * (total number of songs played)) + 100((1/3) * (total number of songs played)) = total earnings from royalties
Simplifying:
(100/3) * (total number of songs played) + (100/3) * (total number of songs played) = total earnings from royalties
(200/3) * (total number of songs played) = total earnings from royalties
Given that the total earnings from royalties was $800, we can set up the equation:
(200/3) * (total number of songs played) = 800
Solving for the total number of songs played:
total number of songs played = (3/200) * 800
total number of songs played = 12
Finally, substitute this value back into the equations for x and y:
x = (2/3) * 12 = 8
y = (1/3) * 12 = 4
Therefore, the solution to the system of equations is (x, y) = (8, 4). Gianna's songs were played on 8 commercials and 4 movies.
can you give me the coordinates to plot
Apologies for the confusion earlier. The correct coordinates to plot are:
(x, y) = (8, 4)
The x-coordinate represents the number of commercials played (8) and the y-coordinate represents the number of movies played (4).
can you give the line coordinates that intercept
Certainly! The line coordinates that intercept are:
The x-intercept is when the y-coordinate is 0:
(x, y) = (8, 0)
The y-intercept is when the x-coordinate is 0:
(x, y) = (0, 4)
These points represent the intersections of the line with the x-axis (x-intercept) and y-axis (y-intercept).
give me the coordinates the intersect (8,4)
I apologize for the confusion. The coordinates you requested are the coordinates of the point of intersection between the line and the axes.
The coordinates of the point of intersection with the x-axis (x-intercept) are:
(x, y) = (8, 0)
The coordinates of the point of intersection with the y-axis (y-intercept) are:
(x, y) = (0, 4)
These points represent the intersections of the line with the x-axis and y-axis. The given point (8, 4) does not represent an intersection with the axes. I apologize for any misunderstanding caused.
To solve the problem graphically, we need to set up a system of equations based on the given information.
Let's assume that the number of commercials Gianna's songs were played on is represented by 'x', and the number of movies is represented by 'y'.
From the given information, we know that Gianna earns $50 every time her song is played in a commercial. So, the total earnings from commercials would be 50x.
Similarly, Gianna earns $100 every time her song is played in a movie. So, the total earnings from movies would be 100y.
We also know that the total earnings from both commercials and movies combined is $800. So, we can set up the equation:
50x + 100y = 800
Now, we are given that the number of commercials is twice the number of movies. Mathematically, this can be written as:
x = 2y
Now, we have a system of equations:
50x + 100y = 800
x = 2y
To graphically solve this system, we can plot the equations on a graph and find the point of intersection.
Let's start by rewriting the second equation in terms of x:
x - 2y = 0
Now, we can graph both equations on the same graph:
Equation 1: 50x + 100y = 800
Equation 2: x - 2y = 0
To plot these equations, we need to rearrange them in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
For equation 1, rearranging gives:
100y = -50x + 800
y = (-1/2)x + 8
For equation 2, rearranging gives:
2y = x
y = (1/2)x
Now, we can plot these lines on a graph and find the point of intersection, which represents the number of commercials and movies played.
Using the graph, we can determine the coordinates of the point of intersection, which represent the values of x and y.