Write a system of equations to describe the situation below, solve using elimination, and fill in the blanks. $\newline$ A boutique in Salem specializes in leather goods for men. Last month, the company sold $93$ wallets and $33$ belts, for a total of $\$6,063$ . This month, they sold $16$ wallets and $13$ belts, for a total of $\$1,336$ . How much does the boutique charge for each item? $\newline$ The boutique charges $\$\_\_\_\_$ for a wallet, and $\$\_\_\_\_$ for a belt.

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Solve a system of equations using elimination: word problems

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

\newline

A boutique in Salem specializes in leather goods for men. Last month, the company sold

93

wallets and

33

belts, for a total of

\$6,063

. This month, they sold

16

wallets and

13

belts, for a total of

\$1,336

. How much does the boutique charge for each item?

\newline

The boutique charges

\$\_\_\_\_

for a wallet, and

\$\_\_\_\_

for a belt.

Define Variables: Let's denote the price of a wallet as $W$ and the price of a belt as $B$ . We can then write two equations based on the information given for the two months. $\newline$ For last month: $93W + 33B = 6063$ $\newline$ For this month: $16W + 13B = 1336$
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 $W$ or $B$ the same as in the first equation. Let's choose to eliminate $W$ by multiplying the second equation by $\frac{93}{16}$ , which is the coefficient of $W$ in the first equation divided by the coefficient of $W$ in the second equation.
Multiply Second Equation: Multiplying the second equation by $\frac{93}{16}$ gives us: $\newline$ $\left(\frac{93}{16} \times 16W\right) + \left(\frac{93}{16} \times 13B\right) = \frac{93}{16} \times 1336$ $\newline$ This simplifies to: $\newline$ $93W + \left(\frac{93}{16} \times 13B\right) = 7803$
Subtract Equations: Now we have two equations with the same coefficient for $W$ : $93W + 33B = 6063$ $93W + \left(\frac{93}{16} \times 13B\right) = 7803$ We can now subtract the first equation from the second to eliminate $W$ .
Combine Like Terms: Subtracting the first equation from the second gives us: $\newline$ $93W + \left(\frac{93}{16} \times 13B\right) - \left(93W + 33B\right) = 7803 - 6063$ $\newline$ This simplifies to: $\newline$ $\left(\frac{93}{16} \times 13B\right) - 33B = 1740$
Convert to Common Denominator: To combine like terms, we need a common denominator. The common denominator for $16$ and $1$ (since $33B$ is the same as $\frac{33}{1} \times B$ ) is $16$ . So we convert $33B$ to $(33 \times \frac{16}{16})B$ .
Solve for B: Now we have: $\newline$ $(\frac{93}{16} \times 13B) - (33 \times \frac{16}{16})B = 1740$ $\newline$ This simplifies to: $\newline$ $(\frac{1209}{16})B - (\frac{528}{16})B = 1740$
Calculate Value of B: Combining the B terms gives us: $\newline$ $(\frac{1209}{16})B - (\frac{528}{16})B = (\frac{681}{16})B$ $\newline$ So we have: $\newline$ $(\frac{681}{16})B = 1740$
Round to Nearest Whole Number: To solve for $B$ , we multiply both sides by the reciprocal of $\frac{681}{16}$ , which is $\frac{16}{681}$ : $B = 1740 \times \left(\frac{16}{681}\right)$
Round to Nearest Whole Number: To solve for $B$ , we multiply both sides by the reciprocal of $\frac{681}{16}$ , which is $\frac{16}{681}$ :
$B = 1740 \times \left(\frac{16}{681}\right)$ Calculating the value of $B$ gives us:
$B = 1740 \times \left(\frac{16}{681}\right) = 41.12$
Since the price of a belt cannot be in cents, we round to the nearest whole number, which gives us $B = \$41$ .