Find the solution of the following system of equations using Gaussian elimination method and show the steps. x-y+z = 4

1. Break Down the Problem

We need to solve the system of equations given by:

1. $x - y + z = 4$

Since there is only one equation with three variables, we cannot find a unique solution. Instead, we can express one variable in terms of the others.

2. Relevant Concepts

For a system of equations, the Gaussian elimination method typically transforms the system into an upper triangular form. Since we only have one equation, we will express one variable in terms of the others.

3. Analysis and Detail

Starting from the equation: $x - y + z = 4$ We can isolate $x$: $x = y - z + 4$

This equation shows that $x$ can take on infinitely many values depending on the choices of $y$ and $z$.

4. Verify and Summarize

Since the original equation contained three variables and only one equation, there are infinite solutions along the line defined by the equation $x = y - z + 4$.

Final Answer

The solution set for the system is: $x = y - z + 4 \quad \text{for any } y, z \in \mathbb{R}$