# Solving a Linear System for Two Tank Salt Solution Problem

• scienceman2k9
In summary, the problem involves two tanks with a constant volume of 500 gallons each, where one tank has pure water pouring in at a rate of 5 gallons per second and salt solution pouring out of the bottom at the same rate into the tank below. The second tank has a drain at the bottom where the solution flows out at a rate of 5 gallons per second. At t=0, the first tank contains 100 lbs of salt and the second tank contains 0 lbs. The amount of salt in each tank over time is represented by x1(t) and x2(t), respectively. To find the expressions for x'1 and x'2, the author has attempted to solve a matrix, but is unsure if
scienceman2k9
For this problem I don't think I am setting up the linear system right:

Two tanks containing 500 gal of salt solution. Pure water pours into the top tank @ 5gal/s. Salt solution pours out of the bottom of the tank and into the tank bellow @ 5 gal/s. There is a drain @ the bottom of the second tank, out of which the solution flows @ a rate of 5 gal/s. @ t=0 there is 100 lbs of salt present in the first tank and zero pounds in the tank immediately below.

Ok, so obviously the volume in the tanks is constant. also, if i let x1(t) and x2(t) be the amount of salt in the respective tanks...X1(0)=100 x2(0)=0

Now I need x'1 and x'2...which I think i may be doing wrong.

I have: x'1=(-5/500)x1 and x'2=(5/500)x1-(5/500)x2

If i factor out a 1/500 and solve the matrix I get a single eigenvalue of -5...but when I find the nullspace I get the v1 and v2 equal to zero which makes no sense. I was thinking that the negative sign on the x'1 equation might not be right, but if I drop that sign and solve the matrix again, I get complex eigenvalues.

Any guidance would be great.

You didn't post a question, just a scenario. What exactly are you trying to find?

an expression for the content of salt in each tank over time, since tis an initial value problem

## 1. How do you set up a linear system for a two tank salt solution problem?

To set up a linear system for a two tank salt solution problem, you will need to define the variables and create equations based on the given information. The variables will represent the amount of salt in each tank, and the equations will relate the variables to each other and to the total amount of salt in the system.

## 2. What are the steps for solving a linear system for a two tank salt solution problem?

The steps for solving a linear system for a two tank salt solution problem are as follows:

• 1. Set up the linear system by defining variables and creating equations.
• 2. Use substitution or elimination to solve for one variable in terms of the other.
• 3. Substitute the found value into the other equation to solve for the remaining variable.
• 4. Check your solution by plugging it back into the original equations.

## 3. What are the different methods for solving a linear system for a two tank salt solution problem?

The two main methods for solving a linear system for a two tank salt solution problem are substitution and elimination. Substitution involves solving for one variable in terms of the other and then substituting the solution into the other equation to find the remaining variable. Elimination involves adding or subtracting the equations to eliminate one variable and then solving for the remaining variable.

## 4. Can you use a calculator to solve a linear system for a two tank salt solution problem?

Yes, you can use a calculator to solve a linear system for a two tank salt solution problem. Many scientific and graphing calculators have built-in functions for solving systems of equations. However, it is important to understand the steps and concepts behind solving a linear system in order to use a calculator effectively.

## 5. Are there any real-life applications of solving a linear system for a two tank salt solution problem?

Yes, there are many real-life applications of solving a linear system for a two tank salt solution problem. For example, this type of problem can be used to calculate the concentration of a salt solution in different tanks or containers, or to determine the amount of salt needed to create a certain concentration in a solution. It can also be applied to other types of mixtures, such as chemicals or ingredients in a recipe.

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