Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Warren is a shoe salesman, and he works on commission. This we... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Warren is a shoe salesman, and he works on commission. This week, there is a special incentive to sell shoes and boots by a certain company. Yesterday, Warren sold 7 pairs of shoes and 1 pair of boots, earning $83 in commission. Today, he sold 1 pair of boots, earning a total commission of $20. How much does Warren earn for the sale of each type of footwear? Warren earns $___ for each pair of shoes and $___ for each pair of boots.
Understand the Problem
The question is asking to create a system of equations based on a real-world scenario involving sales commissions. We need to determine how much Warren earns for each pair of shoes and each pair of boots by setting up and solving equations using an augmented matrix.
Answer
Warren earns $9 for each pair of shoes and $20 for each pair of boots.
Answer for screen readers
Warren earns $9 for each pair of shoes and $20 for each pair of boots.
Steps to Solve
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Identify variables Let ( x ) be the commission Warren earns for each pair of shoes and ( y ) be the commission he earns for each pair of boots.
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Set up equations From the problem, we can establish the following two equations based on Warren's sales:
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From yesterday's sales:
- Warren sold 7 pairs of shoes and 1 pair of boots for a total of $83.
- This gives us the equation: $$ 7x + 1y = 83 $$
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From today's sales:
- Warren sold 1 pair of boots for a total commission of $20.
- This gives us the equation: $$ 0x + 1y = 20 $$
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Write the augmented matrix The system of equations can be represented in augmented matrix form as follows: $$ \begin{bmatrix} 7 & 1 & | & 83 \ 0 & 1 & | & 20 \end{bmatrix} $$
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Use row operations to solve the matrix First, we can simplify the second row. We already have ( y = 20 ).
Next, substitute ( y ) back into the first equation:
- Substitute: $$ 7x + 1(20) = 83 $$
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Solve for x This simplifies to: $$ 7x + 20 = 83 $$ Subtract 20 from both sides: $$ 7x = 63 $$ Then divide by 7: $$ x = 9 $$
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Conclusion Now we have ( x = 9 ) and ( y = 20 ). Thus, Warren earns $9 for each pair of shoes and $20 for each pair of boots.
Warren earns $9 for each pair of shoes and $20 for each pair of boots.
More Information
Warren's commission reflects typical earnings in a sales commission structure, where earnings vary based on the type and quantity of products sold. This problem illustrates how to translate real-world scenarios into algebraic expressions for solving systems of equations.
Tips
- Misinterpreting the totals: Ensure both equations accurately reflect the sales and total commissions.
- Forgetting to simplify the augmented matrix: Always ensure to perform the necessary row operations to simplify before substituting values.
- Incorrectly solving for a variable: Pay close attention to algebraic steps, especially during substitution or isolation of variables.
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