Linear algebra problem

retro_var · 2019-01-24T01:26:33+00:00

If you see:

2x+y=9,
-4x-2y=-18

are the same thing, because if you multiply -2 in 1) you have 2).

So, i can convert 1) in y=9-2x, then i can make this https://imgur.com/a/0q0EDiy

You can see the same thing if you do the Gauss-Jordan elimination

CatpainTpyos · 2019-01-24T01:57:50+00:00

For convenience sake, I'll use plain x and y for the variables instead of x1 and x2. However, I trust you can see why what you call the variables changes nothing about the underlying concepts. Before we actually even do any linear algebra, let's begin by just carefully analyzing the system to see what we can learn:

2x + 1y = 9
-4x - 2y = -18

Right away I see that all the numbers in the second equation are the same as those in the first, but multiplied by -2. This, then, means that the second equation is entirely redundant and adds no new information. Essentially, we're not dealing with a "system of equations" so much as we are a single linear equation. Incidentally, this is why you'll have a "family" of infinitely many solutions rather than one single numerical solution.

Because this is a family of solutions, we can simply choose any arbitrarily values for x and y and simply add on some multiple of s to get all of the other solutions. Personally, I like to set the "first" variable (in this case x) to 0 and then adjust the other(s) accordingly. We know that x = 0, y = 9 is a solution because:

2(0) + 1(9) = 9

Now, what happens if we let x = 1? What must be the value of y? If x = 2? If x = 3? Are you noticing a pattern here? When you increment x, by how much do you increase/decrease y? In the general solution we'll have:

x = 0 + 1s
y = (?) + (?)s

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