Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ The owner of two hotels is ordering towels. He bought $15$ hand towels and $93$ bath towels for his hotel in Princeton, spending a total of $\$1,098$ . He also ordered $74$ hand towels and $72$ bath towels for his hotel in Lancaster, spending $\$1,162$ . How much does each towel cost? $\newline$ A hand towel costs $\$\_\_\_\_$ , and a bath towel costs $\$\_\_\_\_\_$ .

View step-by-step help

Home

Math Problems

Precalculus

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

The owner of two hotels is ordering towels. He bought

15

hand towels and

93

bath towels for his hotel in Princeton, spending a total of

\$1,098

. He also ordered

74

hand towels and

72

bath towels for his hotel in Lancaster, spending

\$1,162

. How much does each towel cost?

\newline

A hand towel costs

\$\_\_\_\_

, and a bath towel costs

\$\_\_\_\_\_

Define variables: Let's define two variables: let $x$ be the cost of one hand towel and $y$ be the cost of one bath towel. We can then write two equations based on the information given: $\newline$ $1$ ) For the Princeton hotel: $15x + 93y = 1,098$ $\newline$ $2$ ) For the Lancaster hotel: $74x + 72y = 1,162$
Use elimination method: To use elimination, we need to eliminate one of the variables. We can do this by multiplying the first equation by a number that will make the coefficient of $x$ or $y$ in both equations the same. Let's multiply the first equation by $74$ and the second equation by $15$ , so the coefficients of $x$ will be the same. $\newline$ First equation multiplied by $74$ : $(15x + 93y) \times 74 = 1,098 \times 74$ $\newline$ Second equation multiplied by $15$ : $(74x + 72y) \times 15 = 1,162 \times 15$
Perform multiplication: Now let's perform the multiplication: $\newline$ First equation: $1110x + 6882y = 81,252$ $\newline$ Second equation: $1110x + 1080y = 17,430$
Eliminate variable x: Next, we subtract the second equation from the first to eliminate x: $\newline$ $(1110x + 6882y) - (1110x + 1080y) = 81,252 - 17,430$ $\newline$ This simplifies to: $\newline$ $6882y - 1080y = 81,252 - 17,430$
Solve for y: Now we solve for y: $\newline$ $5802y = 63,822$ $\newline$ $y = \frac{63,822}{5802}$ $\newline$ $y = 11$ $\newline$ So, each bath towel costs $\$11$ .
Substitute $y$ into equation: Now that we know the cost of each bath towel, we can substitute $y = 11$ into one of the original equations to find $x$ . Let's use the first equation: $\newline$ $15x + 93y = 1,098$ $\newline$ $15x + 93(11) = 1,098$
Solve for x: Now we solve for x: $\newline$ $15x + 1023 = 1,098$ $\newline$ $15x = 1,098 - 1023$ $\newline$ $15x = 75$ $\newline$ $x = \frac{75}{15}$ $\newline$ $x = 5$ $\newline$ So, each hand towel costs $\$5$ .