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.

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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.