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Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.Gabrielle is selling her handmade jewelry online. Yesterday, she sold bracelets and necklaces, for a profit of . Today, she made a profit of by selling bracelets and necklaces. How much profit does Gabrielle earn from each piece?Gabrielle earns a profit of from every bracelet and from every necklace.
Full solution
Q. Write a system of equations to describe the situation below, solve using any method, and fill in the blanks.Gabrielle is selling her handmade jewelry online. Yesterday, she sold bracelets and necklaces, for a profit of . Today, she made a profit of by selling bracelets and necklaces. How much profit does Gabrielle earn from each piece?Gabrielle earns a profit of from every bracelet and from every necklace.
- Define Profit Equations: Let's denote the profit Gabrielle earns from each bracelet as and from each necklace as . The first equation comes from the sales of yesterday: bracelets and necklaces made a profit of $\(212\).\(\newline\)Which equation represents the provided information?\(\newline\)\(4\)b + \(10\)n = \(212\)
- Sales from Yesterday: The second equation comes from today's sales: \(10\) bracelets and \(10\) necklaces made a profit of \(\$260\). Which equation represents the provided information? \(10b + 10n = 260\)
- Sales from Today: System of equations:\(\newline\)\(4b + 10n = 212\)\(\newline\)\(10b + 10n = 260\)\(\newline\)We can solve this system using the method of substitution or elimination. Let's use elimination to solve for one of the variables.
- System of Equations: To eliminate \( n \), we can multiply the first equation by \(-1\) to make the coefficients of \( n \) opposites.\(\newline\)\(-1\) * (\(4\)b + \(10\)n) = \(-1\) * \(212\)\(\newline\)\(-4\)b - \(10\)n = \(-212\)\(\newline\)Now we have:\(\newline\)\(-4\)b - \(10\)n = \(-212\)\(\newline\)\(10\)b + \(10\)n = \(260\)
- Eliminate Variable n: Add the two equations together to eliminate \( n \).\(\newline\)(\(-4\)b - \(10\)n) + (\(10\)b + \(10\)n) = \(-212\) + \(260\)\(\newline\)\(-4\)b - \(10\)n + \(10\)b + \(10\)n = \(48\)\(\newline\)\(6\)b = \(48\)\(\newline\)Now, solve for \( b \).\(\newline\)b = \(48\) / \(6\)\(\newline\)b = \(8\)
- Add Equations: Now that we have the value for \( b \), we can substitute it back into one of the original equations to solve for \( n \). Let's use the first equation:\(\newline\)\(4\)b + \(10\)n = \(212\)\(\newline\)\(4\)(\(8\)) + \(10\)n = \(212\)\(\newline\)\(32\) + \(10\)n = \(212\)\(\newline\)\(10\)n = \(212\) - \(32\)\(\newline\)\(10\)n = \(180\)\(\newline\)n = \(180\) / \(10\)\(\newline\)n = \(18\)
