Solving a consistency problem by Gaussian Elimination

Question

• My Question:

  Textbook solution says that the "plane passes through origin and points (1,0,1), and (0,1,1)". Could you please explain how do we know that? I have a hard time understanding the final solution.

• Textbook Question:

Answer 1

Please note this is the entire problem in the order in which the textbook describes it.

In this problem, we begin with regular Gaussian Elimination.

In this first step, we lay the equation's coefficients in a matrix. Each one corresponding to the proper equation.

In the second step, we manipulate the coefficients so that the second and third rows are equivalent.

In the third step, we multiply equation 3 by -1 to that we can eliminate rows 2 and 3.

In the fourth step eliminate, making the third row equivalent to 0.

So since the third equation looks like 0x₁ + 0x₂ + 0x₃ = b₃ - b₂ - b₁ we know that 0 = b₃ - b₂ - b₁

From here we can pull out -1 leaving us with 0 = -1(-b₃ + b₂ + b₁)

We divide negative one into the 0 leaving us with 0 = -b₃ + b₂ + b₁

Therefore, b₂ + b₁ = b₃

Use the commutative property, you get b₁ + b₂ = b₃

Next, the textbook tells us that the equation must satisfy b₃ = b₁ + b₂ or 0 = b₃ - b₂ - b₁

So we take the first potential solution and put it in a matrix. Essentially, in the matrix, we are saying [b₁ b₂ b₃].

Next, we split the 2 matrices, where their sum is equal to the original.

Then we assume b₁ and b₂ = 1. You may be why this occurs. Don't forget to keep in mind these are random numbers.

This happens because we could get b₃ by adding 1*b₁ + 1*b₂. Therefore it fits the quality 2 = 1+1. Also, it fits the other quality necessary: 0 = b₃ - b₂ - b₁ of 0 = 2 - 1 - 1 or 0 = 0. Therefore, this completely arbitrary point works.