Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ An administrative assistant is making some copies. She made $34$ one-sided copies and $11$ two-sided copies for the V.P. of Marketing, which took a total of $135$ seconds. Next, she made $23$ one-sided copies and $42$ two-sided copies for the Director of Sales, which took $195$ seconds. How long does it take to make each type of copy? $\newline$ It takes $\_$ seconds to make a one-sided copy and $\_$ seconds to make a two-sided copy.

View step-by-step help

Home

Math Problems

Algebra 1

Solve a system of equations using elimination: word problems

Full solution

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

\newline

An administrative assistant is making some copies. She made

34

one-sided copies and

11

two-sided copies for the V.P. of Marketing, which took a total of

135

seconds. Next, she made

23

one-sided copies and

42

two-sided copies for the Director of Sales, which took

195

seconds. How long does it take to make each type of copy?

\newline

It takes

\_

seconds to make a one-sided copy and

\_

seconds to make a two-sided copy.

Define variables: Define the variables for the time it takes to make each type of copy. $\newline$ Let $x$ be the time (in seconds) it takes to make a one-sided copy. $\newline$ Let $y$ be the time (in seconds) it takes to make a two-sided copy.
Write equations: Write the system of equations based on the given information. $\newline$ For the V.P. of Marketing: $\newline$ $34$ one-sided copies and $11$ two-sided copies took $135$ seconds. $\newline$ $34x + 11y = 135$ $\newline$ For the Director of Sales: $\newline$ $23$ one-sided copies and $42$ two-sided copies took $195$ seconds. $\newline$ $23x + 42y = 195$
Multiply first equation: Multiply the first equation by a number that will allow us to eliminate one of the variables when we subtract the equations. $\newline$ We can multiply the first equation by $2$ to align the coefficients of $y$ with the second equation. $\newline$ $(34x + 11y) \times 2 = 135 \times 2$ $\newline$ $68x + 22y = 270$
Subtract equations: Now we have the system of equations: $\newline$ $68x + 22y = 270$ $\newline$ $23x + 42y = 195$ $\newline$ We will subtract the second equation from the first to eliminate $y$ . $\newline$ $(68x + 22y) - (23x + 42y) = 270 - 195$ $\newline$ $68x + 22y - 23x - 42y = 270 - 195$
Solve for x: Perform the subtraction to solve for x. $\newline$ $68x - 23x + 22y - 42y = 270 - 195$ $\newline$ $45x - 20y = 75$
Multiply second equation: We need to find a suitable multiple of the second original equation to eliminate $x$ . We can multiply the second original equation by $2$ to align the coefficients of $x$ with the new equation we have. $(23x + 42y) \times 2 = 195 \times 2$ $46x + 84y = 390$
Subtract equations: Now we have the system of equations: $\newline$ $45x - 20y = 75$ $\newline$ $46x + 84y = 390$ $\newline$ We will subtract the first equation from the second to eliminate $x$ . $\newline$ $(46x + 84y) - (45x - 20y) = 390 - 75$ $\newline$ $46x - 45x + 84y + 20y = 315$
Solve for $y$ : Perform the subtraction to solve for $y$ . $46x - 45x + 84y + 20y = 390 - 75$ $x + 104y = 315$ Since we have only $1x$ , we made a mistake in our calculations. We need to correct this.