Algebra and Trigonometry (MindTap Course List)
4th Edition
ISBN:9781305071742
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter10: Systems Of Equations And Inequalities
Section10.1: Systems Of Linear Equations In Two Variables
Problem 69E
Related questions
Question
![### Mixture Problem: Alcohol Solutions
**Question**: How many liters of 5% alcohol solution and 9% alcohol solution must be mixed to obtain 200 liters of a 7% alcohol solution?
This problem requires determining the quantities of two different alcohol solutions that need to be mixed together to create a new solution with a specific concentration. Here, we aim to mix a 5% alcohol solution with a 9% alcohol solution to get 200 liters of a 7% alcohol solution.
### Steps to Solve the Problem
1. **Set up the Variables**:
- Let \( x \) be the amount (in liters) of the 5% alcohol solution.
- Let \( y \) be the amount (in liters) of the 9% alcohol solution.
2. **Total Volume Equation**:
- The sum of the volumes of the two solutions should be equal to the total desired volume of 200 liters.
\[
x + y = 200
\]
3. **Alcohol Content Equation**:
- The sum of the pure alcohol from both solutions should equal the pure alcohol in the final mixture.
- For the 5% solution: \( 0.05x \)
- For the 9% solution: \( 0.09y \)
We need the mixture to be 200 liters of 7% alcohol:
\[
0.05x + 0.09y = 0.07 \times 200
\]
4. **Solve the System of Equations**:
\[
\begin{cases}
x + y = 200 \\
0.05x + 0.09y = 14
\end{cases}
\]
5. **Substitute and Solve**:
- From the first equation: \( y = 200 - x \)
- Substitute \( y \) in the second equation:
\[
0.05x + 0.09(200 - x) = 14
\]
6. **Simplify and Solve for \( x \)**:
\[
0.05x + 18 - 0.09x = 14
\]
\[
-0.04x + 18 = 14
\]
\[
-0.04x = -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70fe4f99-9e69-4bef-afe6-80d54e8903ee%2F2cf1d5c1-38b5-4b83-aa1b-8a2e7f12c03e%2Fu95b7db_processed.png&w=3840&q=75)
Transcribed Image Text:### Mixture Problem: Alcohol Solutions
**Question**: How many liters of 5% alcohol solution and 9% alcohol solution must be mixed to obtain 200 liters of a 7% alcohol solution?
This problem requires determining the quantities of two different alcohol solutions that need to be mixed together to create a new solution with a specific concentration. Here, we aim to mix a 5% alcohol solution with a 9% alcohol solution to get 200 liters of a 7% alcohol solution.
### Steps to Solve the Problem
1. **Set up the Variables**:
- Let \( x \) be the amount (in liters) of the 5% alcohol solution.
- Let \( y \) be the amount (in liters) of the 9% alcohol solution.
2. **Total Volume Equation**:
- The sum of the volumes of the two solutions should be equal to the total desired volume of 200 liters.
\[
x + y = 200
\]
3. **Alcohol Content Equation**:
- The sum of the pure alcohol from both solutions should equal the pure alcohol in the final mixture.
- For the 5% solution: \( 0.05x \)
- For the 9% solution: \( 0.09y \)
We need the mixture to be 200 liters of 7% alcohol:
\[
0.05x + 0.09y = 0.07 \times 200
\]
4. **Solve the System of Equations**:
\[
\begin{cases}
x + y = 200 \\
0.05x + 0.09y = 14
\end{cases}
\]
5. **Substitute and Solve**:
- From the first equation: \( y = 200 - x \)
- Substitute \( y \) in the second equation:
\[
0.05x + 0.09(200 - x) = 14
\]
6. **Simplify and Solve for \( x \)**:
\[
0.05x + 18 - 0.09x = 14
\]
\[
-0.04x + 18 = 14
\]
\[
-0.04x = -
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