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Solution to augmented matrix? Never seen a problem like this one. How do I solve it?

Solution to augmented matrix? Never seen a problem like this one. How do I solve it-example-1
User Dung Ngo
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The row of zeros at the bottom immediately lets us conclude "infinitely many solutions". This system is consistent and dependent. This is because we have 0x+0y+0z = 0 aka 0 = 0 which is always true for any choice of x, y, and z.

The second row of values lead to the equation 0x+0y+1z = 6, aka z = 6

The first row says: 1x+6y+0z = 1 which is the same as x+6y = 1

Let's say we isolate x

x+6y = 1

x+6y-6y = 1-6y

x = 1-6y

Then we have these three values or expressions

  • x = 1-6y
  • y = any real number
  • z = 6

All of the infinitely many solutions are of the form (x,y,z) = (1-6y, y, 6)

A lot of textbooks will use a parameter such as t, so we could write it as (1-6t, t, 6).

The choice of the letter for the parameter does not matter. I think t is most popular because it represents time. Each (x,y,z) point could represent a particle's location at any given time.

  • If t = 0, then we have (1, 0, 6)
  • If t = 1, then we have (-5, 1, 6)
  • If t = 2, then we have (-11, 2, 6)
  • If t = 3, then we have (-17, 3, 6)

and so on.

=============================

Conclusion:

There are infinitely many solutions of the form (x,y,z) = (1-6t, t, 6) where t is any real number.

This system is consistent and dependent.

User Efosa
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