Understanding Solutions to Systems of Linear Equations

In summary, the given system of equations can be represented as a matrix and solved by taking the free variable to be z. This leads to the general solution of x=0 and y=0, with z as the free variable. The solutions for x and y can also be obtained by taking them as the free variable, resulting in different general solutions. However, in this case, taking z as the free variable leads to the correct solution for the system.
  • #1
roam
1,271
12

Homework Statement



This is from a solved problem:

Here is a system:

[tex]\begin{bmatrix} {1 & 0&0 \\ -1&0&0\\3&-5&0 \end{bmatrix} \begin{bmatrix} {x \\ y\\z \end{bmatrix}= \begin{bmatrix} {0\\ 0\\0\end{bmatrix}[/tex]

A general solution of this system is

[tex]x= \begin{bmatrix} {0 \\ 0\\t \end{bmatrix}= \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} t[/tex]

So, how did they get this solution? I tried solving this and I just don't get it!

The Attempt at a Solution



The matrix corresponds to the set of equations:

x=0
-x=0
3x-5y=0

If I take "y" to be the free variable, I have x=5/3y therefore the solution to the system will be

[tex]\begin{bmatrix} {x \\ y\\z \end{bmatrix} = \begin{bmatrix} {5/3 \\ 1\\0 \end{bmatrix} y + \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} z[/tex]

But if I take "x" to be the free variable I get y=3/5x, so I get the following solution:

[tex]\begin{bmatrix} {x \\ y\\z \end{bmatrix} = \begin{bmatrix} {1 \\ 3/5\\0 \end{bmatrix} x + \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} z[/tex]

How do I know which one (x or y) should be taken as the free variable?

Could anyone please explain to me how to get the right answer? :frown: I just can't see how to deal with this type of problems.
 
Physics news on Phys.org
  • #2
roam said:

Homework Statement



This is from a solved problem:



So, how did they get this solution? I tried solving this and I just don't get it!

The Attempt at a Solution



The matrix corresponds to the set of equations:

x=0
-x=0
3x-5y=0
From these equations, x is obviously 0. With x = 0, the 3rd equation implies that y = 0. This means that z is the free variable.
roam said:
If I take "y" to be the free variable, I have x=5/3y therefore the solution to the system will be

[tex]\begin{bmatrix} {x \\ y\\z \end{bmatrix} = \begin{bmatrix} {5/3 \\ 1\\0 \end{bmatrix} y + \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} z[/tex]

But if I take "x" to be the free variable I get y=3/5x, so I get the following solution:

[tex]\begin{bmatrix} {x \\ y\\z \end{bmatrix} = \begin{bmatrix} {1 \\ 3/5\\0 \end{bmatrix} x + \begin{bmatrix} {0 \\ 0\\1 \end{bmatrix} z[/tex]

How do I know which one (x or y) should be taken as the free variable?

Could anyone please explain to me how to get the right answer? :frown: I just can't see how to deal with this type of problems.
 

Related to Understanding Solutions to Systems of Linear Equations

1. What is a system of linear equations?

A system of linear equations is a set of two or more equations with two or more unknown variables. These equations are all linear, meaning that the highest power of the variable is 1.

2. How do you solve a system of linear equations?

There are several methods for solving a system of linear equations, including substitution, elimination, and graphing. These methods involve manipulating the equations to isolate one of the variables, and then using that value to solve for the other variables.

3. What is the significance of the solution to a system of linear equations?

The solution to a system of linear equations represents the point where all of the equations intersect. This means that the values of the variables in the solution satisfy all of the equations in the system.

4. Can a system of linear equations have more than one solution?

Yes, a system of linear equations can have one, zero, or infinitely many solutions. The number of solutions depends on the number of equations in the system and the relationship between them.

5. How is a system of linear equations used in real life?

Systems of linear equations are used in many real-life applications, such as solving for unknown quantities in physics and engineering problems, optimizing production and cost in business, and determining the intersection of lines in navigation and mapping. They are also essential in data analysis and machine learning algorithms.

Similar threads

Linear independence of Coordinate vectors as columns & rows
    • Calculus and Beyond Homework Help
Replies
2
Views
1K
Avoid unpleasant integrals in solving IVP
    • Calculus and Beyond Homework Help
Replies
3
Views
613
Finding the largest ellipse that will fit in a polygon
    • Calculus and Beyond Homework Help
Replies
4
Views
2K
Codomain and Range of Linear Transformation
    • Calculus and Beyond Homework Help
Replies
10
Views
2K
If ##A^3=0## then ##A-Z## is nonsingular
    • Calculus and Beyond Homework Help
Replies
28
Views
1K
Linear Algebra: LU Decomposition
    • Calculus and Beyond Homework Help
Replies
6
Views
1K
Find the basis so that the matrix will be diagonal
    • Calculus and Beyond Homework Help
Replies
14
Views
2K
One-Dimensional Wave Equation & Steady-State Temperature Distribution
    • Calculus and Beyond Homework Help
Replies
14
Views
1K
How do I know if my eigenvectors are right?
    • Calculus and Beyond Homework Help
Replies
12
Views
1K
[LinAlg] Find the dimension of the subspace
    • Calculus and Beyond Homework Help
Replies
12
Views
2K

Hot Threads

Recent Insights

Back
Top