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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineThe owner of two hotels is ordering towels. He bought 100100 hand towels and 3434 bath towels for his hotel in Oak Grove, spending a total of $672\$672. He also ordered 6262 hand towels and 4545 bath towels for his hotel in Summerfield, spending $608\$608. How much does each towel cost?\newlineA hand towel costs $\$_____, and a bath towel costs $\$_____.

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Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.\newlineThe owner of two hotels is ordering towels. He bought 100100 hand towels and 3434 bath towels for his hotel in Oak Grove, spending a total of $672\$672. He also ordered 6262 hand towels and 4545 bath towels for his hotel in Summerfield, spending $608\$608. How much does each towel cost?\newlineA hand towel costs $\$_____, and a bath towel costs $\$_____.
  1. Define Cost Equations: Let's denote the cost of a hand towel as xx dollars and the cost of a bath towel as yy dollars. The owner bought 100100 hand towels and 3434 bath towels for the Oak Grove hotel, spending a total of $672\$672. This gives us the first equation:\newline100x+34y=672100x + 34y = 672
  2. Summerfield Hotel Purchase: For the Summerfield hotel, the owner bought 6262 hand towels and 4545 bath towels, spending a total of $608\$608. This gives us the second equation:\newline62x+45y=60862x + 45y = 608
  3. Solve Using Elimination: We now have a system of two equations with two variables:\newline100x+34y=672100x + 34y = 672\newline62x+45y=60862x + 45y = 608\newlineWe can solve this system using either substitution or elimination. Let's use the elimination method to solve for one of the variables.
  4. Multiply Equations: To eliminate one of the variables, we can multiply the second equation by a number that will make the coefficient of xx in the second equation equal to the coefficient of xx in the first equation when subtracted. We can multiply the second equation by 100100 and the first equation by 6262:(100)(62x+45y)=(100)(608)(100)(62x + 45y) = (100)(608)(62)(100x+34y)=(62)(672)(62)(100x + 34y) = (62)(672)
  5. Subtract Equations: After multiplying, we get:\newline6200x+4500y=608006200x + 4500y = 60800\newline6200x+2108y=417446200x + 2108y = 41744\newlineNow, we subtract the second equation from the first to eliminate xx:\newline(6200x+4500y)−(6200x+2108y)=60800−41744(6200x + 4500y) - (6200x + 2108y) = 60800 - 41744
  6. Solve for yy: Perform the subtraction:\newline6200x+4500y−6200x−2108y=60800−417446200x + 4500y - 6200x - 2108y = 60800 - 41744\newline4500y−2108y=60800−417444500y - 2108y = 60800 - 41744\newline2392y=190562392y = 19056
  7. Substitute to Find x: Now, we solve for yy by dividing both sides of the equation by 23922392:y=190562392y = \frac{19056}{2392}y=8y = 8So, each bath towel costs $8\$8.
  8. Substitute to Find x: Now, we solve for yy by dividing both sides of the equation by 23922392:y=190562392y = \frac{19056}{2392}y=8y = 8So, each bath towel costs $8\$8.Now that we have the value of yy, we can substitute it back into one of the original equations to solve for xx. Let's use the first equation:100x+34y=672100x + 34y = 672100x+34(8)=672100x + 34(8) = 672
  9. Substitute to Find x: Now, we solve for yy by dividing both sides of the equation by 23922392:y=190562392y = \frac{19056}{2392}y=8y = 8So, each bath towel costs $8\$8.Now that we have the value of yy, we can substitute it back into one of the original equations to solve for xx. Let's use the first equation:100x+34y=672100x + 34y = 672100x+34(8)=672100x + 34(8) = 672Simplify the equation and solve for xx:100x+272=672100x + 272 = 672100x=672−272100x = 672 - 272100x=400100x = 400x=400100x = \frac{400}{100}x=4x = 4So, each hand towel costs $4\$4.

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