Write a system of equations to describe the situation below, solve using any method, and fill in the blanks. A boutique in Booneville specializes in leather goods for men. Last month, the company sold 60 wallets and 22 belts, for a total of $3,422. This month, they sold 81 wallets and 22 belts, for a total of $4,073. How much does the boutique charge for each item? The boutique charges $_____ for a wallet, and $_____ for a belt.

Question

Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.  A boutique in Booneville specializes in leather goods for men. Last month, the company sold 60 wallets and  22 belts, for a total of $3,422. This month, they sold 81 wallets and  22 belts, for a total of $4,073. How much does the boutique charge for each item?  The boutique charges $_____ for a wallet, and $_____ for a belt.

Bytelearn · Accepted Answer

Define Variables: Let's denote the price of a wallet as $ w $ and the price of a belt as $ b $. We are given that 60 wallets and 22 belts were sold for a total of $3,422 last month. This can be represented by the equation: $$ 60w + 22b = 3422 $$Given Sales Data: We are also given that this month, they sold 81 wallets and 22 belts for a total of $4,073. This can be represented by the equation: $$ 81w + 22b = 4073 $$System of Equations: We now have a system of equations to solve: $$ \begin{cases} 60w + 22b = 3422 \ 81w + 22b = 4073 \end{cases} $$ To solve this system, we can use the method of elimination or substitution. Let's use elimination to solve for one of the variables.Elimination Method: To eliminate $ b $, we can multiply the first equation by -1 and add it to the second equation: $$ -1 \times (60w + 22b) = -1 \times 3422 $$ $$ -60w - 22b = -3422 $$ $$ 81w + 22b = 4073 $$ Adding the two equations: $$ (81w + 22b) + (-60w - 22b) = 4073 + (-3422) $$ $$ 21w = 651 $$Solve for Wallet Price: Now we can solve for $ w $: $$ 21w = 651 $$ $$ w = \frac{651}{21} $$ $$ w = 31 $$ So, the boutique charges $31 for a wallet.Substitute Wallet Price: Now that we have the price of a wallet, we can substitute $ w = 31 $ into one of the original equations to find $ b $. Let's use the first equation: $$ 60(31) + 22b = 3422 $$ $$ 1860 + 22b = 3422 $$ $$ 22b = 3422 - 1860 $$ $$ 22b = 1562 $$Solve for Belt Price: Now we can solve for $ b $: $$ 22b = 1562 $$ $$ b = \frac{1562}{22} $$ $$ b = 71 $$ So, the boutique charges $71 for a belt.

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