Write the solution set of the given homogeneous system in parametric vector form. x1+3x2−5x3=0 x1+4x2−8x3=0 −3x1−7x2+9x3=0

Question

Write the solution set of the given homogeneous system in parametric vector form.  x1+3x2−5x3=0  x1+4x2−8x3=0  −3x1−7x2+9x3=0

Forneadil · Accepted Answer

The given system is  x1+3x2−5x3=0  x1+4x2−8x3=0  −3x1−7x2+9x3=0  The augmented matrix is  [13−5014−80−3−790]  Reduce matrix to row echelon form  [−3−790053−50023−20]R3−2/5R2→R33/5R2→R2∼[−3−79001−300000]  [−3−79001−300000]R1+7R2→R1−1/3R1→R1[104001−300000]  Thus we get  x1+4x3=0⇒x1=−4x3  x2−3x3=0⇒x2=3x3  Hence, the given homogeneous system in parametric vector form is  x=[x1x2x3]=x3[−431]

RizerMix · Accepted Answer

Step 1:The augmented matrix for the system is:[13−5|014−8|0−3−79|0]Performing row operations:[13−5|001−3|002−6|0](R2&amp;nbsp;=&amp;nbsp;R2&amp;nbsp;-&amp;nbsp;R1)[13−5|001−3|0000|0](R3&amp;nbsp;=&amp;nbsp;R3&amp;nbsp;-&amp;nbsp;2R2)Step 2:Next, we can express the row-echelon form of the augmented matrix as a system of equations:x1+3x2−5x3=0x2−3x3=00=0Simplifying the equations, we have:x1+3x2−5x3=0x2=3x30=0Step 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=−4tTherefore, the solution set in parametric vector form is:𝐱=[x1x2x3]=[−4t3tt]wheret∈ℝNote that the parameter t represents any real number and can take any value, which will give different solutions to the homogeneous system.

Vasquez · Accepted Answer

Result:[x1x2x3]=[−3t3tt],where&amp;nbsp;t∈ℝ.Solution:The given homogeneous system can be written in matrix form as:[13−514−8−3−79][x1x2x3]=[000]Using Gaussian elimination or row reduction, we can find the row echelon form of the augmented matrix:[13−5|014−8|0−3−79|0]→[13−5|001−3|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&#039;s assign the parameter t to x3. Then we can express x1 and x2 in terms of t:x1=−3tx2=3tx3=tTherefore, the solution set of the given homogeneous system in parametric vector form is:[x1x2x3]=[−3t3tt],where&amp;nbsp;t∈ℝ.

Don Sumner · Accepted Answer

To solve the given homogeneous system of equations:x1+3x2−5x3=0x1+4x2−8x3=0−3x1−7x2+9x3=0we can represent it in augmented matrix form and perform row operations to find the solution set.[13−5014−80−3−790]Applying row operations to the augmented matrix, we can transform it to row-echelon form:[13−5001−300000]From this row-echelon form, we can deduce that:x1=−3x2+5x3x2=3x3Therefore, 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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