Answer to Question #220179 in Algebra for Rashmay

Question

 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=pt 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? 

Answer 1

The three equations created

𝑥+2𝑦−𝑧=5

2𝑥+3𝑦−3𝑧=-3

𝑥+𝑦+𝑧=0

Row Echelon Matrix

Row \space Operation \space 1:

$Row \space Operation \space 1\\ \begin{bmatrix} 1 & 2 & -1 & 5 \\ 2 & 3 & -3 & -3 \\ 1 & 1 & 1 & 0 \end{bmatrix} Add \space 2 \space times \space the \space 1st \space row\space to \space the \space 2nd \space row \space \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & -1 & -1 & -13 \\ 1 & 1 & 1 & 0 \end{bmatrix}\\ Row \space Operation \space 2:\\ \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & -1 & -1 & -13 \\ 1 & 1 & 1 & 0 \end{bmatrix} Add \space -1 \space times \space the \space 1st \space row\space to \space the \space 3rd \space row \space \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & -1 & -1 & -13 \\ 0 & -1 & 2 & -5 \end{bmatrix}\\ Row \space Operation \space 3:\\ \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & -1 & -1 & -13 \\ 0 & -1 & 2 & -5 \end{bmatrix} Multiply \space the \space 2nd \space row \space by \space -1 \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 1 & 13 \\ 0 & -1 & 2 & -5 \end{bmatrix}\\ Row \space Operation \space 4:\\ \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & -1 & -1 & -13 \\ 0 & -1 & 2 & -5 \end{bmatrix} Add \space 1 \space times \space2 the \space 2nd \space row \space to \space the \space 3rd \space row \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 1 & 13 \\ 0 & 0 & 3 & 8 \end{bmatrix}\\ Row \space Operation \space 5:\\ \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 1 & 13 \\ 0 & 0 & 3 & 8 \end{bmatrix} Multiply \space the \space 3rd \space row \space by \space 1/3 \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 1 & 13 \\ 0 & 0 & 1 & 8/3 \end{bmatrix}\\ Row \space Operation \space 6:\\ \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 1 & 13 \\ 0 & 0 & 1 & 8/3 \end{bmatrix} Add \space -1 \space times \space2 the \space 3rd \space row \space to \space the \space 2nd \space row \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 0 & 31/3 \\ 0 & 0 & 1 & 8/3 \end{bmatrix}\\ Row \space Operation \space 7:\\ \begin{bmatrix} 1 & 2 & -1 & 5 \\ 0 & 1 & 0 & 31/3 \\ 0 & 0 & 1 & 8/3 \end{bmatrix} Add \space 1 \space times \space2 the \space 3rd \space row \space to \space the \space 1st \space row \begin{bmatrix} 1 & 2 & 0 & 23/3 \\ 0 & 1 & 0 & 31/3 \\ 0 & 0 & 1 & 8/3 \end{bmatrix}\\ Row \space Operation \space 8:\\ \space\\ \begin{bmatrix} 1 & 2 & 0 & 23/3 \\ 0 & 1 & 0 & 31/3 \\ 0 & 0 & 1 & 8/3 \end{bmatrix} Add \space 1 \space times \space2 the \space 3rd \space row \space to \space the \space 1st \space row \begin{bmatrix} 1 & 0 & 0 & -13 \\ 0 & 1 & 0 & 31/3 \\ 0 & 0 & 1 & 8/3 \end{bmatrix}\\$

Therefore the solution of the three equations is as follows;

x= -31

y= 31/3

z= 8/3

Selected Task

2. What did the graphs show when you entered the second equation? 

When I entered the second equation the graph showed two planes meet each on a line of intersection.