Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Tisha is a salon owner. Yesterday, she did 4 haircuts and colo... Write a system of equations to describe the situation below, solve using an augmented matrix, and fill in the blanks. Tisha is a salon owner. Yesterday, she did 4 haircuts and colored the hair of 5 clients, charging a total of $643. Today, she colored the hair of 1 client, charging a total of $99. How much does Tisha charge for her services? Tisha charges $ for a haircut and $ for a coloring.
Understand the Problem
The question is asking to set up a system of equations based on a scenario involving Tisha's hair salon services and to solve it using an augmented matrix. We need to figure out how much Tisha charges for a haircut and for a coloring service.
Answer
Tisha charges $37 for a haircut and $99 for a coloring.
Answer for screen readers
Tisha charges $37 for a haircut and $99 for a coloring.
Steps to Solve
- Define Variables
Let ( x ) represent the cost of a haircut and ( y ) represent the cost of a coloring service.
- Set Up Equations Based on the Problem
From the information provided:
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For the first scenario (yesterday):
- Tisha did 4 haircuts and 5 colorings for a total of $643. This gives us the equation: $$ 4x + 5y = 643 $$
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For the second scenario (today):
- Tisha did 1 coloring for a total of $99, which gives us the equation: $$ 0x + 1y = 99 $$
- Write the Augmented Matrix
The system of equations can be represented in augmented matrix form as follows: $$ \begin{bmatrix} 4 & 5 & | & 643 \ 0 & 1 & | & 99 \end{bmatrix} $$
- Use Row Reduction to Solve the Matrix
Apply row operations to convert the matrix to reduced row echelon form:
- Start with: $$ \begin{bmatrix} 4 & 5 & | & 643 \ 0 & 1 & | & 99 \end{bmatrix} $$
- To simplify the first row, divide by 4: $$ \begin{bmatrix} 1 & \frac{5}{4} & | & \frac{643}{4} \ 0 & 1 & | & 99 \end{bmatrix} $$
- Now eliminate ( y ) from the first row by substituting ( y = 99 ): $$ 1x + \frac{5}{4}(99) = \frac{643}{4} $$ Calculating gives: $$ 1x + \frac{495}{4} = \frac{643}{4} $$
- Subtract ( \frac{495}{4} ) from both sides: $$ x = \frac{643}{4} - \frac{495}{4} = \frac{148}{4} = 37 $$
- Find the Values of ( x ) and ( y )
Now substitute ( x = 37 ) back into the equation for ( y ): $$ y = 99 $$
Now we have:
- Cost of a haircut ( x = 37 )
- Cost of a coloring service ( y = 99 )
Tisha charges $37 for a haircut and $99 for a coloring.
More Information
This problem involves setting up a system of equations based on real-life scenarios. The process of using an augmented matrix simplifies solving linear equations. Tisha's pricing allows us to calculate costs effectively by representing the services mathematically.
Tips
- Forgetting to label the variables can lead to confusion about which value corresponds to which service.
- Incorrectly simplifying the rows when using the augmented matrix can lead to incorrect values.
- Miscalculating the total when substituting back into the equations can skew results.
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