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Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineA boutique in Salem specializes in leather goods for men. Last month, the company sold 9393 wallets and 3333 belts, for a total of $6,063\$6,063. This month, they sold 1616 wallets and 1313 belts, for a total of $1,336\$1,336. How much does the boutique charge for each item?\newlineThe boutique charges $____\$\_\_\_\_ for a wallet, and $____\$\_\_\_\_ for a belt.

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Q. Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks.\newlineA boutique in Salem specializes in leather goods for men. Last month, the company sold 9393 wallets and 3333 belts, for a total of $6,063\$6,063. This month, they sold 1616 wallets and 1313 belts, for a total of $1,336\$1,336. How much does the boutique charge for each item?\newlineThe boutique charges $____\$\_\_\_\_ for a wallet, and $____\$\_\_\_\_ for a belt.
  1. Define Variables: Let's denote the price of a wallet as WW and the price of a belt as BB. We can then write two equations based on the information given for the two months.\newlineFor last month: 93W+33B=606393W + 33B = 6063\newlineFor this month: 16W+13B=133616W + 13B = 1336
  2. Eliminate Variable: To use elimination, we need to eliminate one of the variables. We can do this by multiplying the second equation by a number that will make the coefficient of WW or BB the same as in the first equation. Let's choose to eliminate WW by multiplying the second equation by 9316\frac{93}{16}, which is the coefficient of WW in the first equation divided by the coefficient of WW in the second equation.
  3. Multiply Second Equation: Multiplying the second equation by 9316\frac{93}{16} gives us:\newline(9316×16W)+(9316×13B)=9316×1336\left(\frac{93}{16} \times 16W\right) + \left(\frac{93}{16} \times 13B\right) = \frac{93}{16} \times 1336\newlineThis simplifies to:\newline93W+(9316×13B)=780393W + \left(\frac{93}{16} \times 13B\right) = 7803
  4. Subtract Equations: Now we have two equations with the same coefficient for WW:93W+33B=606393W + 33B = 606393W+(9316×13B)=780393W + \left(\frac{93}{16} \times 13B\right) = 7803We can now subtract the first equation from the second to eliminate WW.
  5. Combine Like Terms: Subtracting the first equation from the second gives us:\newline93W+(9316×13B)−(93W+33B)=7803−606393W + \left(\frac{93}{16} \times 13B\right) - \left(93W + 33B\right) = 7803 - 6063\newlineThis simplifies to:\newline(9316×13B)−33B=1740\left(\frac{93}{16} \times 13B\right) - 33B = 1740
  6. Convert to Common Denominator: To combine like terms, we need a common denominator. The common denominator for 1616 and 11 (since 33B33B is the same as 331×B\frac{33}{1} \times B) is 1616. So we convert 33B33B to (33×1616)B(33 \times \frac{16}{16})B.
  7. Solve for B: Now we have:\newline(9316×13B)−(33×1616)B=1740(\frac{93}{16} \times 13B) - (33 \times \frac{16}{16})B = 1740\newlineThis simplifies to:\newline(120916)B−(52816)B=1740(\frac{1209}{16})B - (\frac{528}{16})B = 1740
  8. Calculate Value of B: Combining the B terms gives us:\newline(120916)B−(52816)B=(68116)B(\frac{1209}{16})B - (\frac{528}{16})B = (\frac{681}{16})B\newlineSo we have:\newline(68116)B=1740(\frac{681}{16})B = 1740
  9. Round to Nearest Whole Number: To solve for BB, we multiply both sides by the reciprocal of 68116\frac{681}{16}, which is 16681\frac{16}{681}:B=1740×(16681)B = 1740 \times \left(\frac{16}{681}\right)
  10. Round to Nearest Whole Number: To solve for BB, we multiply both sides by the reciprocal of 68116\frac{681}{16}, which is 16681\frac{16}{681}:
    B=1740×(16681)B = 1740 \times \left(\frac{16}{681}\right)Calculating the value of BB gives us:
    B=1740×(16681)=41.12B = 1740 \times \left(\frac{16}{681}\right) = 41.12
    Since the price of a belt cannot be in cents, we round to the nearest whole number, which gives us B=$41B = \$41.

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