Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Warren earns $___ for each pair of shoes and $___ for each pai... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Warren earns $___ for each pair of shoes and $___ for each pair of boots.
Understand the Problem
The question is asking us to set up a system of equations based on Warren's sales data and then solve it using an augmented matrix. Specifically, we need to determine the commission Warren earns for each pair of shoes and each pair of boots.
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
- Define the Variables
Let $s$ represent the commission earned per pair of shoes, and let $b$ represent the commission earned per pair of boots.
- Set Up the Equations
From the problem, we can derive the following equations:
-
For yesterday's sales:
- Warren sold 7 pairs of shoes and 1 pair of boots, earning a total of $83.
- This gives us the equation: $$ 7s + 1b = 83 $$
-
For today's sales:
- Warren sold 1 pair of boots, earning a total of $20.
- This gives us the equation: $$ 0s + 1b = 20 $$
- Write the Augmented Matrix
Next, we can express the system of equations in augmented matrix form: $$ \begin{bmatrix} 7 & 1 & | & 83 \ 0 & 1 & | & 20 \end{bmatrix} $$
- Use Row Reduction
We will use row reduction to solve the matrix. Start with the matrix: $$ \begin{bmatrix} 7 & 1 & | & 83 \ 0 & 1 & | & 20 \end{bmatrix} $$
We can replace the first row using the second row to eliminate $b$ from the first equation:
- Multiply the second row by 1 and subtract it from the first row:
$$ R_1 = R_1 - (1 \times R_2) $$
The new matrix will be: $$ \begin{bmatrix} 7 & 0 & | & 63 \ 0 & 1 & | & 20 \end{bmatrix} $$
- Back Substitute
Now, from the second row, we can conclude: $$ b = 20 $$
From the first row: $$ 7s = 63 $$ Thus: $$ s = \frac{63}{7} = 9 $$
- Final Results
We have found: $$ s = 9 $$ $$ b = 20 $$
This means 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 earnings reflect the typical commission-based sales model, where higher sales volume can significantly affect overall commissions. In this case, he consistently sold more shoes than boots, which influenced his earnings per item sold.
Tips
- Forgetting to define variables clearly can lead to confusion later in the process.
- Incorrectly setting up the equations from the word problem may lead to wrong results. Always double-check the alignment of quantities with their respective earnings.
- Misapplying the row reduction steps can cause errors in the final calculation.
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