Write a system of equations to describe the situation below, solve using elimination, 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 $2$ sitcoms and $5$ dramas, for a total of $290$ minutes of programming, not counting commercials. On Tuesday nights, he has scheduled $4$ sitcoms and $1$ drama, for a total of $130$ 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.

Full solution

Q. Write a system of equations to describe the situation below, solve using elimination, 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

2

sitcoms and

5

dramas, for a total of

290

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

4

sitcoms and

1

drama, for a total of

130

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.

Define variables: Define the variables for the lengths of the sitcoms and dramas. $\newline$ Let $x$ be the length of one sitcom in minutes. $\newline$ Let $y$ be the length of one drama in minutes.
Write Monday equation: Write the equation for Monday nights. $\newline$ $2$ sitcoms and $5$ dramas take up $290$ minutes. $\newline$ $2x + 5y = 290$
Write Tuesday equation: Write the equation for Tuesday nights. $\newline$ $4$ sitcoms and $1$ drama take up $130$ minutes. $\newline$ $4x + y = 130$
Decide variable to eliminate: Decide which variable to eliminate. We can eliminate $y$ by multiplying the second equation by $-5$ 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.\[2x + 5y = 290\]\[2(20) + 5(50) = 290\]\[40 + 250 = 290\]\[290 = 290\]