John is a songwriter who collects royalties on his songs whenever they are played in a commercial or a movie. John will earn $20 every time one of his songs is played in a commerci... John is a songwriter who collects royalties on his songs whenever they are played in a commercial or a movie. John will earn $20 every time one of his songs is played in a commercial and $80 every time one is played in a movie. John's songs were played on 4 times as many commercials as movies, and he earned a total of $320. Graphically solve a system of equations to determine the number of commercials, x, and the number of movies, y, on which John's songs were played.
Understand the Problem
The question is asking to set up a system of equations based on the given information about John's earnings from commercials and movies, then solve it graphically to find the number of commercials and movies.
Answer
John's songs were played on 8 commercials and 2 movies.
Answer for screen readers
The number of commercials, $x$, is 8, and the number of movies, $y$, is 2.
Steps to Solve
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Define Variables Let $x$ be the number of commercials and $y$ be the number of movies.
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Set Up Equations From the problem:
- John earns $20 per commercial and $80 per movie, leading to the equation: $$ 20x + 80y = 320 $$
- It's given that he played songs 4 times as often in commercials as in movies, which can be represented as: $$ x = 4y $$
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Substitute for y Substituting $x = 4y$ into the earnings equation: $$ 20(4y) + 80y = 320 $$
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Combine Like Terms Simplifying the equation: $$ 80y + 80y = 320 $$ $$ 160y = 320 $$
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Solve for y Dividing both sides by 160: $$ y = \frac{320}{160} = 2 $$
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Find x Using the value of $y$ to find $x$: $$ x = 4y = 4(2) = 8 $$
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Graph the Equations To solve graphically, plot the lines for:
- $20x + 80y = 320$
- $x = 4y$
- Identify the Intersection Point The graph will show the intersection at the point $(8, 2)$, which represents the number of commercials and movies.
The number of commercials, $x$, is 8, and the number of movies, $y$, is 2.
More Information
John made a total of $320 from 8 commercials and 2 movies. This setup demonstrates how to create and solve a system of linear equations both algebraically and graphically.
Tips
- Confusing the relationship: Remember that $x = 4y$, not the other way around.
- Miscalculating total earnings while setting up the equation: Ensure to clearly understand how much is earned per commercial and per movie.
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