Discussion Assignment
Your discussion response for this unit will consist of two parts.
First, create 3 equations of the form $ax+by+cz=d$, where $a,b,c$, and $d$ are constants (integers between -5 and 5). For example, $x+2y-z=-1$. Perform row operations on your system to obtain a row-echelon form and the solution.
Go to the 3D calculator website GeoGebra: www.geogebra.org/3d?lang Ept and enter each of the equations.

After you have completed this first task, choose one of the following to complete your discussion post.
1. Reflect on what the graphs are suggesting for one equation, two equations, and three equations, and describe your observations. Think about the equation as a function $f$ of $x$ and $y$, for example, $x+2y+1=z$ in the example above. Geogebra automatically interprets this way, that is, like $z=f(x,y)=x+2y+1$, it isolates $z$ in the equation.
2. What did the graphs show when you entered the second equation?

Step 1 ：Let's create three equations of the form $ax+by+cz=d$, where $a,b,c$, and $d$ are constants (integers between -5 and 5). For example, we can have the following equations:

Step 2 ：$$x+2y-z=-1$$

Step 3 ：$$2x-y+z=2$$

Step 4 ：$$x-2y+z=-1$$

Step 5 ：We can represent these equations in matrix form as follows:

Step 6 ：$$\left[\begin{array}{ccc}1& 2& -1\\ 2& -1& 1\\ 1& -2& 1\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}-1\\ 2\\ -1\end{array}\right]$$

Step 7 ：We can perform row operations on this matrix to obtain a row-echelon form and find the solution to the system of equations.

Step 8 ：After performing the row operations, we find that the solution to the system of equations is $x=-1$, $y=4$, and $z=8$.

Step 9 ：Each of these equations represents a plane in the 3D space. The solution to the system of equations is the intersection of these planes. If the planes intersect at a single point, the system has a unique solution. If the planes intersect along a line, the system has infinitely many solutions. If the planes do not intersect, the system has no solution.

Step 10 ：This geometric interpretation can be visualized using the GeoGebra 3D calculator.

Step 11 ：Final Answer: The solution to the system of equations is $\boxed{{\displaystyle -1,4,8}}$.