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Section 2.7 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 2.7.1. 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 2.7.2. Three planes intersecting at a single point
Example 2.7.3. 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 2.7.4. Three planes intersecting in a line
Example 2.7.5. 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 2.7.6. Three pairwise intersecting planes with no common point
Example 2.7.7. 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 2.7.8. Three parallel planes