Linear Algebra General Solution

In summary, the conversation is discussing finding the general solution to a system of equations with variables a, b, c, and d. The solution involves solving for y by dividing both sides by ad - bc, which is possible because it is stated that ad - bc != 0. The general solution will result in a pair of numbers for x and y, and it can be checked by substituting into the original equations. It is noted that a and c cannot both be zero.
  • #1
Precursor
222
0
Homework Statement
Find the general solution to the system:
[tex]ax+ by= 1[/tex]
[tex]cx+ dy= 2[/tex]

Consider the case when
[tex]ad- bc \neq 0[/tex]

The attempt at a solution
Like in my other post, I multiplied the first equation by "c" and the second equation by "a", and then I subtracted the two equations. I just seem to be stuck.

I got the following matrix:
[tex]0...ad- bc...2a- c[/tex]
[tex]ac...ad...2a[/tex]

Therefore, [tex](ad- bc)y= 2a- c[/tex]
But what relation can I deduce from this equation to help my determine a general solution for the system? The previous question I posted was much more obvious, where I was able to solve y= 0. But this isn't the case. I appreciate any help.
 
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  • #2
Solve for y by dividing both sides by ad - bc. You know that it is legitimate to do this because you are told that ad - bc != 0.

Just as before, the general solution is going to be just a single point - a pair of numbers. The reason that a unique solution exists, although it might not seem obvious to you at this point in your studies, is that ad - bc != 0.
 
  • #3
Mark44 said:
Solve for y by dividing both sides by ad - bc. You know that it is legitimate to do this because you are told that ad - bc != 0.

Just as before, the general solution is going to be just a single point - a pair of numbers. The reason that a unique solution exists, although it might not seem obvious to you at this point in your studies, is that ad - bc != 0.

When I divide both sides by y, I get y= (2a- c)/(ad- bc)
If I substitute this back into either one of the original equations, I get an even more complicated equation. Is there some way to simply it?
 
  • #4
What you get for x will be no more complicated than what you got for y, at least when it is simplified. Also, you can check your answers by substituting them into your original system of equations. It should be true that ax + by = 1 and cx + dy = 2.
 
  • #5
When I substitute what I got for y into the first original equation, I get ax= 1- b((2a-c)/(ad-bc))
Can I simplify this?
 
  • #6
Yes. Get a common denominator and things simplify.

Also, as in the other problem you posted, a and c can't both be zero, due to the restriction that ad - bc != 0.
 

FAQ: Linear Algebra General Solution

What is the general solution in linear algebra?

The general solution in linear algebra is a set of all possible solutions to a system of linear equations. It is a combination of the particular solution and the homogeneous solution. The particular solution represents the specific values for the variables that satisfy the equations, while the homogeneous solution represents the values that make the equations equal to zero.

How do you find the general solution in linear algebra?

To find the general solution in linear algebra, you first need to solve the system of linear equations by using methods such as Gaussian elimination or substitution. Once you have the solution for the particular solution, you can then find the homogeneous solution by setting all the variables to zero and solving for the remaining variables. The general solution is then represented as a linear combination of the particular solution and the homogeneous solution.

Why is the general solution important in linear algebra?

The general solution is important in linear algebra because it provides a complete and comprehensive understanding of the solutions to a system of linear equations. It not only gives the specific values for the variables but also the range of values that satisfy the equations. This allows for a deeper understanding of the relationships between the variables and how they affect each other.

What is the difference between a particular solution and a general solution in linear algebra?

A particular solution in linear algebra is a specific set of values for the variables that satisfies the system of equations. It is unique and represents one solution to the equations. On the other hand, the general solution is a set of all possible solutions that includes the particular solution and the homogeneous solution. The general solution provides a more comprehensive understanding of the solutions to the equations.

How do you check if a solution is a general solution in linear algebra?

To check if a solution is a general solution in linear algebra, you can substitute the values for the variables into the original system of equations. If the equations are satisfied with these values, then it is a general solution. Another way to check is to use the properties of linear combinations. A general solution is represented as a linear combination of the particular solution and the homogeneous solution, so it should satisfy the properties of linear combinations.

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