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Describe and compare the solution sets of $x_{1}-3 x_{2}+5 x_{3}=0$ and $x_{1}-3 x_{2}+5 x_{3}=4$

$X=x_{2}\left[\begin{array}{l}{3} \\ {1} \\ {0}\end{array}\right]+x_{3}\left[\begin{array}{c}{-5} \\ {0} \\ {1}\end{array}\right]=x_{2} u+x_{3} v$

The solution set of the homogeneous equation is the plane through the origin in $R^{3}$ spanned by u and $v$ . The solution set of the non homogeneous equation is parallel to this plane and passes through the point $p=\left[\begin{array}{l}{4} \\ {0} \\ {0}\end{array}\right]$

Algebra

Chapter 1

Linear Equations in Linear Algebra

Section 5

Solution Sets of Linear Systems

Introduction to Matrices

Campbell University

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University of Michigan - Ann Arbor

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all right. So in these problem, we were given two equations and that we need to compare the solution. Says how these two equations. So we first have x one, minus three, x two thanks to plus five x three, which is zero. So we can who who tow we can put x x two terms and x three terms. Um, on the right hand side. So that is X one because the X two minus five x three So that means we have a social set that x one determined by X to the next three and extra and excrete are three variables here. That's three x two minus five x three and x to the next three. Now we separate these factors by, uh bye bye. The terms that contain X to annex three. So we first to check out the terms of containing X juice with three ex too x two and zero plus next five x three zero and next three. So that is our first solution. You compare Measure the vector for so that is 310 time sex too. And next, five 01 times x three. All right, So for their first equal for the second equation. Here we have X one minus three x two I was five x three equals four. Now again, we put a three x two and five x three on the right hand side, so that is X one can be returned. Informs off ex to a next three. So that's three x two minus five x three plus four. So this ocean said this three x to minus five x three plus four and next to x three are three variables. Now we separate this vector that is three x two x 20 and active five x three X That should be zero and next three last term. He's accounts tent, which is for you to do now we take out a skater, so I will Right down here. Now we take out a skater. So that is three one zoo times x two and next five 01 times x three and the council in term is or 00 So that that is our solution for the 2nd 1 So we could compare these two solutions. We can actually find out that the only difference is that the second solution has accounts in term here that it's because because we have Ah, we have, Ah, homogeneous least time here. So we have a council for my hand side up the equality. But Porter Percy, for the first equation, we don't have such a such a council way only have zero. So that is a homogeneous in your system. So we can running down as, ah, as a parametric direct victor forms without any constant better. But the second equation things are different. So that's something we need to be careful.

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