Macros Chemical Company is trying produce a test batch of a new organic fertilizer made from a proprietary type of green algae. The fertilizer is an experimental mixture of 4 ingedients: salt, water, soil, and algae (all measured in liters). In order to remain stable, the ratio of salt to water must be 2 to 10. In addition, the number of liters of salt and soil combined must always be the same as the number of liters of algae and water combined. How many liters of each compound should be included to create 130 liters of the fertilizer? Set up a system of equations, and then solve using your calculator. Use the variables in order in your equations as x, y, z, w as shown below in part a.

Solve the system; write the solutions here, using 
t as the parameter as needed. Solution:   
 
x=    liters of salt ;
 
  y = liters of water ;
 
 z=   liters of soil ; and
 
  w = liters of algae.


To solve this problem, we need to create a system of equations based on the given conditions. Let's define the variables:

- \( x \): liters of salt
- \( y \): liters of water
- \( z \): liters of soil
- \( w \): liters of algae

### Step 1: Set up the system of equations
From the problem, we know the following:

1. **Salt to water ratio**: The ratio of salt to water must be \( 2:10 \). This means:
   \[
   \frac{x}{y} = \frac{2}{10} \quad \text{or equivalently} \quad 10x = 2y
   \]
   which simplifies to:
   \[
   5x = y \tag{1}
   \]

2. **Liters of salt and soil combined must equal liters of algae and water combined**:
   \[
   x + z = y + w \tag{2}
   \]

3. **The total amount of fertilizer is 130 liters**: The sum of all the components must add up to 130 liters:
   \[
   x + y + z + w = 130 \tag{3}
   \]

### Step 2: Express the system of equations

We now have three equations:

1. \( 5x = y \)
2. \( x + z = y + w \)
3. \( x + y + z + w = 130 \)

### Step 3: Solve the system of equations

#### Substituting equation (1) into equations (2) and (3):

From equation (1), we have \( y = 5x \). Substitute this into equations (2) and (3):

- From equation (2): 
  \[
  x + z = 5x + w \quad \text{or} \quad z = 4x + w \tag{4}
  \]

- From equation (3): 
  \[
  x + 5x + z + w = 130 \quad \text{or} \quad 6x + z + w = 130 \tag{5}
  \]

#### Substitute equation (4) into equation (5):
Substitute \( z = 4x + w \) into equation (5):
\[
6x + (4x + w) + w = 130
\]
Simplify:
\[
10x + 2w = 130
\]
Solve for \( w \):
\[
2w = 130 - 10x \quad \text{or} \quad w = 65 - 5x \tag{6}
\]

#### Final substitutions:

Now that we have \( w = 65 - 5x \), substitute this into equation (4) to find \( z \):
\[
z = 4x + (65 - 5x) = 65 - x \tag{7}
\]

### Step 4: Solutions in terms of \( x \):

- \( y = 5x \)
- \( w = 65 - 5x \)
- \( z = 65 - x \)

### Step 5: Parameterized solution:

Since \( x \) (the amount of salt) can be any value that satisfies these conditions, the solution is:

\[
x = t \quad (\text{where } t \text{ is a parameter}),
\]
\[
y = 5t,
\]
\[
z = 65 - t,
\]
\[
w = 65 - 5t.
\]

### Step 6: Conclusion

The solutions for the number of liters of each compound in terms of \( t \) (the parameter) are:

- \( x = t \) liters of salt
- \( y = 5t \) liters of water
- \( z = 65 - t \) liters of soil
- \( w = 65 - 5t \) liters of algae

### Tip:

You can now choose any value for \( t \) to generate a specific solution, keeping in mind that \( t \) must be a positive number such that all variables remain non-negative.

---

Would you like to go over the steps or see examples with specific values for \( t \)?

### Related Questions:
1. What is the smallest value \( t \) can take for all amounts to remain positive?
2. What happens to the amount of algae as \( t \) increases?
3. How does the total amount of salt and soil compare to the amount of algae and water at different values of \( t \)?
4. How can we graph the relationship between salt and the other compounds?
5. Can this system be modified if the ratio between salt and water changes?

Tip: Solving systems of equations using substitution can help simplify complex scenarios.

Macros Chemical Company is trying to produce a test batch of a new organic fertilizer made from a proprietary type of green algae. The fertilizer is an experimental mixture of 4 ingredients: salt, water, soil, and algae (all measured in liters). In order to remain stable, the ratio of salt to water must be 2 to 10. In addition, the number of liters of salt and soil combined must always be the same as the number of liters of algae and water combined. How many liters of each compound should be included to create 130 liters of the fertilizer? Set up a system of equations, and then solve using your calculator. Use the variables in order in your equations as x, y, z, w.

Learn how to solve a system of equations to find the composition of a new organic fertilizer made from salt, water, soil, and algae. This problem involves algebraic concepts, including ratios, proportions, and systems of equations, suitable for Grades 10-12.

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