Write the solution set of the given homogeneous system in

mronjo7n

mronjo7n

Answered question

2021-11-23

Write the solution set of the given homogeneous system in parametric vector form.
x1+3x25x3=0
x1+4x28x3=0
3x17x2+9x3=0

Answer & Explanation

Forneadil

Forneadil

Beginner2021-11-24Added 18 answers

The given system is
x1+3x25x3=0
x1+4x28x3=0
3x17x2+9x3=0
The augmented matrix is
[135014803790]
Reduce matrix to row echelon form
[37900535002320]R32/5R2R33/5R2R2[379001300000]
[379001300000]R1+7R2R11/3R1R1[104001300000]
Thus we get
x1+4x3=0x1=4x3
x23x3=0x2=3x3
Hence, the given homogeneous system in parametric vector form is
x=[x1x2x3]=x3[431]
RizerMix

RizerMix

Expert2023-06-17Added 656 answers

Step 1:
The augmented matrix for the system is:
[135|0148|0379|0]
Performing row operations:
[135|0013|0026|0](R2 = R2 - R1)
[135|0013|0000|0](R3 = R3 - 2R2)
Step 2:
Next, we can express the row-echelon form of the augmented matrix as a system of equations:
x1+3x25x3=0x23x3=00=0
Simplifying the equations, we have:
x1+3x25x3=0x2=3x30=0
Step 3:
Now, we can express the solution set in parametric vector form. Let x3=t, where t is a free parameter. We can then express x2 and x1 in terms of t:
x2=3tx1=3x2+5x3=3(3t)+5t=9t+5t=4t
Therefore, the solution set in parametric vector form is:
𝐱=[x1x2x3]=[4t3tt]wheret
Note that the parameter t represents any real number and can take any value, which will give different solutions to the homogeneous system.
Vasquez

Vasquez

Expert2023-06-17Added 669 answers

Result:
[x1x2x3]=[3t3tt],where t.
Solution:
The given homogeneous system can be written in matrix form as:
[135148379][x1x2x3]=[000]
Using Gaussian elimination or row reduction, we can find the row echelon form of the augmented matrix:
[135|0148|0379|0][135|0013|0000|0]
We have obtained a row of zeros, indicating that the system is consistent and has infinitely many solutions.
To express the solution set in parametric vector form, we can assign a parameter to one of the variables. Let's assign the parameter t to x3. Then we can express x1 and x2 in terms of t:
x1=3tx2=3tx3=t
Therefore, the solution set of the given homogeneous system in parametric vector form is:
[x1x2x3]=[3t3tt],where t.
Don Sumner

Don Sumner

Skilled2023-06-17Added 184 answers

To solve the given homogeneous system of equations:
x1+3x25x3=0x1+4x28x3=03x17x2+9x3=0
we can represent it in augmented matrix form and perform row operations to find the solution set.
[135014803790]
Applying row operations to the augmented matrix, we can transform it to row-echelon form:
[135001300000]
From this row-echelon form, we can deduce that:
x1=3x2+5x3x2=3x3
Therefore, the solution set of the given homogeneous system in parametric vector form is:
𝐱=[x1x2x3]=[3x2+5x33x3x3]=[3x200]+[5x30x3]=x2[300]+x3[501]
where x2 and x3 are arbitrary parameters.

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