Systems of Equations: Question

Question

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: geogebra.org/3d?lang=pt) and enter each of the equations. 

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. 

Answer 1

x + 2y - z = -1

x + y + z = 1

x - y - z = -1

1 2 -1 -1

1 1 1 1

1 -1 -1 -1

Step1: -row2 + row1; -row2 + row3

0 1 -2 -2

1 1 1 1

0 -2 -2 -2

Step 2: row3/-2

0 1 -2 -2

1 1 1 1

0 1 1 1

step 3: -row3 + row1

0 0 -3 -3

1 1 1 1

0 1 1 1

step 4: row1/-3

0 0 1 1

1 1 1 1

0 1 1 1

step 4: -row1 + row3; -row1 + row2

0 0 1 1

1 1 0 0

0 1 0 0

step 5: -row3 + row2

0 0 1 1

1 0 0 0

0 1 0 0

The reduced row echelon form is:

1 0 0 0

0 1 0 0

0 0 1 1

The solution is x=0, y=0, z=1

which is where these 3 plane shall intersect