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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ A TV station executive is planning the new lineup for next season's shows. On Monday nights, there will be $6$ sitcoms and $5$ dramas, for a total of $416$ minutes of programming, not counting commercials. On Tuesday nights, he has scheduled $5$ sitcoms and $5$ dramas, for a total of $390$ 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? $\newline$ Sitcoms are $\_$ minutes long and dramas are $\_$ minutes long.

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Solve a system of equations using any method: word problems

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.

\newline

A TV station executive is planning the new lineup for next season's shows. On Monday nights, there will be

6

sitcoms and

5

dramas, for a total of

416

minutes of programming, not counting commercials. On Tuesday nights, he has scheduled

5

sitcoms and

5

dramas, for a total of

390

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?

\newline

Sitcoms are

\_

minutes long and dramas are

\_

minutes long.

Denote Lengths: Let's denote the length of a sitcom as ' $s$ ' minutes and the length of a drama as ' $d$ ' minutes. $\newline$ We need to set up two equations based on the given information.
Equation for Monday: For Monday nights, the total time for $6$ sitcoms and $5$ dramas is $416$ minutes. This gives us the equation: $\newline$ $6s + 5d = 416$
Equation for Tuesday: For Tuesday nights, the total time for $5$ sitcoms and $5$ dramas is $390$ minutes. This gives us the equation: $\newline$ $5s + 5d = 390$
System of Equations: We now have a system of two equations: $\newline$ $6s + 5d = 416$ $\newline$ $5s + 5d = 390$ $\newline$ We can solve this system using the method of elimination or substitution. Let's use elimination.
Elimination Method: To eliminate one of the variables, we can subtract the second equation from the first equation: $\newline$ $(6s + 5d) - (5s + 5d) = 416 - 390$ $\newline$ $6s - 5s + 5d - 5d = 26$ $\newline$ $s = 26$
Find 's' Value: Now that we have the value of 's', we can substitute it back into one of the original equations to find 'd'. Let's use the second equation: $\newline$ $5s + 5d = 390$ $\newline$ $5(26) + 5d = 390$ $\newline$ $130 + 5d = 390$
Substitute to Find 'd': Subtract $130$ from both sides of the equation to solve for 'd': $\newline$ $130 + 5d - 130 = 390 - 130$ $\newline$ $5d = 260$ $\newline$ $d = \frac{260}{5}$ $\newline$ $d = 52$

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