Section 1.7 Geometric examples of three equations in three unknowns
A single equation in three unknowns can be interpreted geometrically in 3-dimensional space. An equation
\begin{equation*}
Ax+By+Cz=D
\end{equation*}
has solutions that form a plane as long as at least one of \(A, B, C\) is nonzero. If we look at a system of such linear equations, the solution set is the set of all points lying in all of the corresponding planes.
Example 1.7.1 . Three planes intersecting at a point.
Consider the system of linear equations
\begin{equation*}
\begin{array}{rl}
x+y+z \amp= 2\\
x+2y+z\amp=3\\
x+2y-3z \amp=2
\end{array}
\end{equation*}
The augmented matrix is
\begin{equation*}
\begin{bmatrix}
1\amp1\amp1\amp2\\
1\amp2\amp1\amp3\\
1\amp2\amp-3\amp2
\end{bmatrix}
\end{equation*}
whose reduced row echelon form is
\begin{equation*}
\begin{bmatrix}
1\amp0\amp0\amp\frac34\\
0\amp1\amp0\amp1\\
0\amp0\amp1\amp\frac14
\end{bmatrix}
\end{equation*}
This means that there is a single solution: \((x,y,z)=(\frac34,1,\frac14)\text{.}\) Here are the three planes corresponding to the three equations:
Figure 1.7.2. Three planes intersecting at a single point
Example 1.7.3 . Three planes intersecting in a single line.
Consider the system of linear equations
\begin{equation*}
\begin{array}{rl}
x+y-z \amp= 3\\
x+y+z\amp=1\\
2x+2y \amp=4
\end{array}
\end{equation*}
The augmented matrix is
\begin{equation*}
\begin{bmatrix}
1\amp1\amp-1\amp3\\
1\amp1\amp1\amp1\\
2\amp2\amp0\amp4
\end{bmatrix}
\end{equation*}
whose reduced row echelon form is
\begin{equation*}
\begin{bmatrix}
1\amp1\amp0\amp2\\
0\amp0\amp1\amp-1\\
0\amp0\amp0\amp0
\end{bmatrix}
\end{equation*}
This means \(y=t\text{,}\) \(x=2-t\) and \(z=-1\) so that \((x,y,z)=(2-t,t,-1)\) is a solution for any real number \(t\text{.}\)
Figure 1.7.4. Three planes intersecting in a line
Example 1.7.5 . Three intersecting planes with no solutions.
Consider the system of linear equations
\begin{equation*}
\begin{array}{rl}
2x-y+z \amp= 1\\
x+y+z\amp=2\\
4x+y+3z \amp=3
\end{array}
\end{equation*}
The augmented matrix is
\begin{equation*}
\begin{bmatrix}
2\amp-1\amp1\amp1\\
1\amp1\amp1\amp2\\
4\amp1\amp3\amp3
\end{bmatrix}
\end{equation*}
whose reduced row echelon form is
\begin{equation*}
\begin{bmatrix}
1\amp0\amp\frac23\amp0\\
0\amp1\amp\frac13\amp0\\
0\amp0\amp0\amp1
\end{bmatrix}
\end{equation*}
The last row indicates that there is no solution.
Figure 1.7.6. Three pairwise intersecting planes with no common point
Example 1.7.7 . Three nonintersecting planes with no solutions.
Consider the system of linear equations
\begin{equation*}
\begin{array}{rl} 2x-y+z \amp= 1\\ 2x-y+z \amp= 2\\ 2x-y+z \amp=3
\end{array}
\end{equation*}
The augmented matrix is
\begin{equation*}
\begin{bmatrix} 2\amp-1\amp1\amp1\\ 2\amp-1\amp1\amp2\\
2\amp-1\amp1\amp3 \end{bmatrix}
\end{equation*}
whose reduced row echelon form is
\begin{equation*}
\begin{bmatrix} 1\amp -\frac12\amp\frac12\amp0\\
0\amp0\amp0\amp1\\ 0\amp0\amp0\amp0\\ \end{bmatrix}
\end{equation*}
The middle row indicates that there is no solution.
Figure 1.7.8. Three parallel planes 