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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.A TV station executive is planning the new lineup for next season's shows. On Monday nights, there will be sitcoms and dramas, for a total of minutes of programming, not counting commercials. On Tuesday nights, he has scheduled sitcoms and drama, for a total of minutes of non-commercial programming. All sitcoms have the same length and all dramas have the same length. How long is each type of show?Sitcoms are minutes long and dramas are minutes long.
Full solution
Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.A TV station executive is planning the new lineup for next season's shows. On Monday nights, there will be sitcoms and dramas, for a total of minutes of programming, not counting commercials. On Tuesday nights, he has scheduled sitcoms and drama, for a total of minutes of non-commercial programming. All sitcoms have the same length and all dramas have the same length. How long is each type of show?Sitcoms are minutes long and dramas are minutes long.
- Define variables: Define the variables for the lengths of the sitcoms and dramas.Let be the length of one sitcom in minutes.Let be the length of one drama in minutes.
- Write Monday equation: Write the equation for Monday nights. sitcoms and dramas take up minutes.
- Write Tuesday equation: Write the equation for Tuesday nights. sitcoms and drama take up minutes.
- Decide variable to eliminate: Decide which variable to eliminate. We can eliminate by multiplying the second equation by and adding it to the first equation.
- Multiply second equation: Multiply the second equation by -5＂).\(\newline\$-5(4x + y) = -5(130)\)\(\newline\)\(-20x - 5y = -650\)
- Add equations to eliminate \(y\): Add the new equation to the first equation to eliminate \(y\).
\((2x + 5y) + (-20x - 5y) = 290 + (-650)\)
\(2x - 20x = 290 - 650\)
\(-18x = -360\) - Solve for x: Solve for x.\(\newline\)\(-18x = -360\)\(\newline\)\(x = \frac{-360}{-18}\)\(\newline\)\(x = 20\)
- Substitute \(x\) into second equation: Substitute \(x\) back into the second original equation to solve for \(y\).\[4x + y = 130\]\[4(20) + y = 130\]\[80 + y = 130\]\[y = 130 - 80\]\[y = 50\]
- Check solution: Check the solution by substituting \(x\) and \(y\) into the first original equation.\[2x + 5y = 290\]\[2(20) + 5(50) = 290\]\[40 + 250 = 290\]\[290 = 290\]
