A chemical supply company has a 30% sulfuric acid solution and a 70% sulfuric acid solu- tion. How many liters of each should be used to ob- tain 100 liters of a 40% solution? 


We can solve this problem using a system of linear equations. Let's define the following variables:

- Let \( x \) represent the number of liters of the 30% sulfuric acid solution.
- Let \( y \) represent the number of liters of the 70% sulfuric acid solution.

We are asked to create a total of 100 liters of a 40% sulfuric acid solution, so we have the following two conditions:

1. The total volume of the mixture must be 100 liters:
   \[
   x + y = 100
   \]

2. The total amount of sulfuric acid in the mixture should give a 40% solution. Since the 30% solution contains \( 0.30x \) liters of sulfuric acid and the 70% solution contains \( 0.70y \) liters of sulfuric acid, the total amount of sulfuric acid in the final 100 liters of the mixture should be \( 0.40 \times 100 = 40 \) liters. Thus, the second equation is:
   \[
   0.30x + 0.70y = 40
   \]

Now, we can solve this system of equations.

### Step 1: Solve the first equation for one variable
From the first equation:
\[
x + y = 100 \quad \Rightarrow \quad y = 100 - x
\]

### Step 2: Substitute this expression into the second equation
Substitute \( y = 100 - x \) into the second equation:
\[
0.30x + 0.70(100 - x) = 40
\]

### Step 3: Simplify and solve for \( x \)
\[
0.30x + 0.70(100 - x) = 40
\]
\[
0.30x + 70 - 0.70x = 40
\]
\[
-0.40x + 70 = 40
\]
\[
-0.40x = 40 - 70
\]
\[
-0.40x = -30
\]
\[
x = \frac{-30}{-0.40} = 75
\]

So, \( x = 75 \) liters of the 30% solution.

### Step 4: Find \( y \)
Since \( y = 100 - x \), we substitute \( x = 75 \):
\[
y = 100 - 75 = 25
\]

### Conclusion
You need 75 liters of the 30% sulfuric acid solution and 25 liters of the 70% sulfuric acid solution to obtain 100 liters of a 40% sulfuric acid solution.

Would you like more details or further explanation on any step?

Here are some related questions:

1. What happens if the desired concentration is 50% instead of 40%?
2. How would the result change if you needed 200 liters of solution?
3. What is the general form of the system of equations for any final concentration?
4. Can you solve this problem graphically, using a coordinate system?
5. What are other methods to solve systems of equations apart from substitution?

**Tip:** When mixing solutions, it's essential to track both the total volume and the concentration of each component to ensure the correct mixture.

A chemical supply company has a 30% sulfuric acid solution and a 70% sulfuric acid solution. How many liters of each should be used to obtain 100 liters of a 40% solution?

Learn how to mix 30% and 70% sulfuric acid solutions to create 100 liters of a 40% sulfuric acid solution. This step-by-step solution explains how to set up and solve a system of linear equations using the substitution method. Suitable for students in Grades 8-10.

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