Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. $\newline$ The owner of two hotels is ordering towels. He bought $100$ hand towels and $34$ bath towels for his hotel in Oak Grove, spending a total of $\$672$ . He also ordered $62$ hand towels and $45$ bath towels for his hotel in Summerfield, spending $\$608$ . How much does each towel cost? $\newline$ A hand towel costs $\$$ _, and a bath towel costs $\$$ _.

View step-by-step help

Home

Math Problems

Algebra 2

Solve a system of equations using any method: word problems

Full solution

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

\newline

The owner of two hotels is ordering towels. He bought

100

hand towels and

34

bath towels for his hotel in Oak Grove, spending a total of

\$672

. He also ordered

62

hand towels and

45

bath towels for his hotel in Summerfield, spending

\$608

. How much does each towel cost?

\newline

A hand towel costs

\$

_____, and a bath towel costs

\$

_____.

Define Cost Equations: Let's denote the cost of a hand towel as $x$ dollars and the cost of a bath towel as $y$ dollars. The owner bought $100$ hand towels and $34$ bath towels for the Oak Grove hotel, spending a total of $\$672$ . This gives us the first equation: $\newline$ $100x + 34y = 672$
Summerfield Hotel Purchase: For the Summerfield hotel, the owner bought $62$ hand towels and $45$ bath towels, spending a total of $\$608$ . This gives us the second equation: $\newline$ $62x + 45y = 608$
Solve Using Elimination: We now have a system of two equations with two variables: $\newline$ $100x + 34y = 672$ $\newline$ $62x + 45y = 608$ $\newline$ We can solve this system using either substitution or elimination. Let's use the elimination method to solve for one of the variables.
Multiply Equations: To eliminate one of the variables, we can multiply the second equation by a number that will make the coefficient of $x$ in the second equation equal to the coefficient of $x$ in the first equation when subtracted. We can multiply the second equation by $100$ and the first equation by $62$ : $(100)(62x + 45y) = (100)(608)$ $(62)(100x + 34y) = (62)(672)$
Subtract Equations: After multiplying, we get: $\newline$ $6200x + 4500y = 60800$ $\newline$ $6200x + 2108y = 41744$ $\newline$ Now, we subtract the second equation from the first to eliminate $x$ : $\newline$ $(6200x + 4500y) - (6200x + 2108y) = 60800 - 41744$
Solve for $y$ : Perform the subtraction: $\newline$ $6200x + 4500y - 6200x - 2108y = 60800 - 41744$ $\newline$ $4500y - 2108y = 60800 - 41744$ $\newline$ $2392y = 19056$
Substitute to Find x: Now, we solve for $y$ by dividing both sides of the equation by $2392$ : $y = \frac{19056}{2392}$ $y = 8$ So, each bath towel costs $\$8$ .
Substitute to Find x: Now, we solve for $y$ by dividing both sides of the equation by $2392$ : $y = \frac{19056}{2392}$ $y = 8$ So, each bath towel costs $\$8$ .Now that we have the value of $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $100x + 34y = 672$ $100x + 34(8) = 672$
Substitute to Find x: Now, we solve for $y$ by dividing both sides of the equation by $2392$ : $y = \frac{19056}{2392}$ $y = 8$ So, each bath towel costs $\$8$ .Now that we have the value of $y$ , we can substitute it back into one of the original equations to solve for $x$ . Let's use the first equation: $100x + 34y = 672$ $100x + 34(8) = 672$ Simplify the equation and solve for $x$ : $100x + 272 = 672$ $100x = 672 - 272$ $100x = 400$ $x = \frac{400}{100}$ $x = 4$ So, each hand towel costs $\$4$ .