Solving \( m \times n \) Linear System
\( m \times n \) linear system: (\( m \) equations in \( n \) unknowns)
\[ \begin{cases} a_{11} x_1 + a_{12} x_2 + \cdots + a_{1n} x_n = b_1 \\ a_{21} x_1 + a_{22} x_2 + \cdots + a_{2n} x_n = b_2 \\ \vdots \qquad \vdots \qquad\qquad \vdots \\ a_{m1} x_1 + a_{m2} x_2 + \cdots + a_{mn} x_n = b_m \end{cases} \]\( \Updownarrow \) In matrix notation: \( A\vec{X} = \vec{B} \)
\[ \begin{bmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & & & \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix} \]\( \Updownarrow \) In terms of linear combination: (columns of \( A \))
\[ x_1 \begin{bmatrix} a_{11} \\ a_{21} \\ a_{31} \\ \vdots \\ a_{m1} \end{bmatrix} + x_2 \begin{bmatrix} a_{12} \\ a_{22} \\ a_{32} \\ \vdots \\ a_{m2} \end{bmatrix} + \cdots + x_n \begin{bmatrix} a_{1n} \\ a_{2n} \\ \vdots \\ a_{mn} \end{bmatrix} = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_m \end{bmatrix} \]For linear system, one and only one of the following cases can occur:
- there is a unique (ie. only one) solution;
- there are infinitely many solutions;
- there is no solution.
In the case of \( 2 \times 2 \) linear system, the above 3 cases correspond to:
(1)
Figure 1
Case (1) diagram for a 2 by 2 linear system.
Coordinate axes are drawn with vertical and
horizontal arrows. Two blue straight lines cross
each other at a single point near the origin,
marked with a dot. A blue arrow and the label
'only one intersection' point to the crossing
point, illustrating a unique solution.
(2)
Figure 2
Case (2) diagram for a 2 by 2 linear system.
Coordinate axes are drawn with vertical and
horizontal arrows. A red line and a blue line
lie directly on top of one another with positive
slope, appearing as a single overlapping line. A
blue arrow with the label '2 lines coincide'
points to them, illustrating infinitely many
solutions.
(3)
Figure 3
Case (3) diagram for a 2 by 2 linear system.
Coordinate axes are drawn with vertical and
horizontal arrows. Two blue parallel lines with
positive slope run without crossing. A blue
arrow and the label '2 unequal parallel lines,
no intersection' point to them, illustrating no
solution.
Ex1
\[ \begin{cases} 2x + y = -1 \\ x - 3y = 5 \end{cases} \] \[ \left[\begin{array}{cc|c} 2 & 1 & -1 \\ 1 & -3 & 5 \end{array}\right] \rightarrow \left[\begin{array}{cc|c} 1 & -3 & 5 \\ 2 & 1 & -1 \end{array}\right] \rightarrow \left[\begin{array}{cc|c} 1 & -3 & 5 \\ 0 & 7 & -11 \end{array}\right] \]\( 7y = -11 \Rightarrow y = -\frac{11}{7} \)
\( x - 3y = 5 \)
\[ \begin{gathered} x = 5 + 3y = 5 - \frac{33}{7} = \frac{2}{7} \\[0.5em] \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} \frac{2}{7} \\ -\frac{11}{7} \end{pmatrix} \end{gathered} \]
Figure 4
A hand-drawn coordinate graph with a
horizontal x-axis and vertical y-axis drawn
in black. Two blue lines are plotted: one
labeled 2x + y = -1 with a steep negative
slope, and another labeled x - 3y = 5 with a
gentle positive slope crossing toward the
right. The two lines intersect at a single
point in the fourth quadrant, labeled with
the coordinates \( \left(\frac{2}{7},
-\frac{11}{7}\right) \). A red arrow and red
text indicate that the 2 lines intersect at
a unique point.
2 lines intersect at a unique point.
Ex2
\[ \begin{cases} x + 2y + z = 1 \\ 3x + y + 4z = 0 \\ 2x + 2y + 3z = 2 \end{cases} \] \[ \left[\begin{array}{ccc|c} 1 & 2 & 1 & 1 \\ 3 & 1 & 4 & 0 \\ 2 & 2 & 3 & 2 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & 2 & 1 & 1 \\ 0 & -5 & 1 & -3 \\ 0 & -2 & 1 & 0 \end{array}\right] \] \[ \rightarrow \left[\begin{array}{ccc|c} 1 & 2 & 1 & 1 \\ 0 & 1 & -\frac{1}{5} & \frac{3}{5} \\ 0 & 1 & -\frac{1}{2} & 0 \end{array}\right] \rightarrow \left[\begin{array}{ccc|c} 1 & 2 & 1 & 1 \\ 0 & 1 & -\frac{1}{5} & \frac{3}{5} \\ 0 & 0 & -\frac{3}{10} & -\frac{3}{5} \end{array}\right] \]\( z = 2 \)
\( y = \frac{3}{5} + \frac{z}{5} = 1 \)
\( x = 1 - 2y - z = 1 - 2 - 2 = -3 \)
\[ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -3 \\ 1 \\ 2 \end{pmatrix} \]
Figure 5
A three-dimensional sketch showing
coordinate axes labeled x, y, and z. Three
planes drawn in blue cut through the space
and all intersect at a single common point
marked near the origin. A red arrow points
to this intersection point with the label
reading 3 planes intersect at a point. This
illustrates the unique solution of the
system, corresponding to the point where x
equals negative 3, y equals 1, and z equals
2.
3 planes intersect at a point.
Ex3
\( x - 6y = 7 \) \(\longleftarrow\) a line
\( y = s \) (free), \( x = 7 + 6s \)
\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 7 + 6s \\ s \end{pmatrix} = \begin{pmatrix} 7 \\ 0 \end{pmatrix} + s \begin{pmatrix} 6 \\ 1 \end{pmatrix} \]parametric form shift direction of line
Figure 6
A graph on x-y axes with a blue vertical
y-axis and blue horizontal x-axis. A red
straight line with positive slope crosses
the x-axis at the point (7,0), labeled in
red below the axis. A purple horizontal
arrow from the origin points right to (7,0),
labeled with the vector (7 over 0),
representing the shift. From (7,0) a second
purple arrow points up and to the right
along the red line, labeled with the vector
(6 over 1), representing the direction of
the line.
Ex4
\[ \begin{cases} x + 3y = -1 \\ 4x + 12y = -4 \end{cases} \]2 same lines
\[ \left[\begin{array}{cc|c} 1 & 3 & -1 \\ 4 & 12 & -4 \end{array}\right] \rightarrow \left[\begin{array}{cc|c} 1 & 3 & -1 \\ 0 & 0 & 0 \end{array}\right] \]\( y = s \) (free)
(pivot) \( x = -1 - 3y = -1 - 3s \)
\[ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} -1 - 3s \\ s \end{pmatrix} = \begin{pmatrix} -1 \\ 0 \end{pmatrix} + s \begin{pmatrix} 3 \\ 1 \end{pmatrix} \]parametric form shift direction
Figure 7
A hand-drawn coordinate graph with a blue
vertical y-axis and blue horizontal x-axis.
Two red lines lie almost exactly on top of
one another, sloping downward from upper
left to lower right, representing the two
coincident lines of the system. A red point
is marked near the origin at approximately
(-1, 0). Magenta arrows drawn along the
lower line represent the shift vector \(
\begin{pmatrix} -1 \\ 0 \end{pmatrix} \)
starting from the origin, and the direction
vector \( \begin{pmatrix} 3 \\ 1
\end{pmatrix} \) extending along the line to
the upper left. The vectors are labeled in
magenta with \( \begin{pmatrix} 3 \\ 1
\end{pmatrix} \) and \( \begin{pmatrix} -1
\\ 0 \end{pmatrix} \).
2 same lines
Ex5
\[ \begin{cases} x + 3y = -1 \\ 2x + 6y = 3 \end{cases} \] \[ \left[\begin{array}{cc|c} 1 & 3 & -1 \\ 2 & 6 & 3 \end{array}\right] \rightarrow \left[\begin{array}{cc|c} 1 & 3 & -1 \\ 0 & 0 & 5 \end{array}\right] \]no solution
\( 0y = 5 \)
Figure 8
A coordinate plane with a horizontal x-axis
and vertical y-axis drawn in black. Two blue
lines with negative slope run parallel to
each other and do not intersect. The lower
line is labeled \( x + 3y = -1 \) and the
upper line is labeled \( 2x + 6y = 3 \). A
red underlined note reads 2 unequal parallel
lines, no intersection, illustrating that
the system has no solution.
2 unequal parallel lines, no intersection
Ex6
\[ \begin{cases} x + 3y - z = 5 \\ 2x - y + z = 10 \end{cases} \] \[ \left(\begin{array}{ccc|c} 1 & 3 & -1 & 5 \\ 2 & -1 & 1 & 10 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & 3 & -1 & 5 \\ 0 & -7 & 3 & 0 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & 3 & -1 & 5 \\ 0 & 1 & -\frac{3}{7} & 0 \end{array}\right) \]\( z = \alpha \) (free)
\( y = \frac{3}{7} z = \frac{3}{7} \alpha \)
\( x = 5 - 3y + z = 5 - \frac{9}{7}\alpha + \alpha = 5 - \frac{2}{7}\alpha \)
\[ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 5 - \frac{2}{7}\alpha \\ \frac{3}{7}\alpha \\ \alpha \end{pmatrix} = \begin{pmatrix} 5 \\ 0 \\ 0 \end{pmatrix} + \alpha \begin{pmatrix} -\frac{2}{7} \\ \frac{3}{7} \\ 1 \end{pmatrix} \]parametric form shift direction of line
Figure 9
A three-dimensional sketch with x, y, and z
axes. Two large planes drawn in blue
intersect each other, meeting along a common
line. A red line runs through this
intersection representing the line of
solutions. Purple arrows near the origin
mark the shift vector and the direction of
the line, showing the two planes intersect
in a line.
Ex7
\[ \begin{cases} x - y + 2z = 7 \\ 2x - 2y + 4z = 14 \end{cases} \] \[ \left(\begin{array}{ccc|c} 1 & -1 & 2 & 7 \\ 2 & -2 & 4 & 14 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & -1 & 2 & 7 \\ 0 & 0 & 0 & 0 \end{array}\right) \]\( z = \alpha \) (free)
\( y = \beta \) (free)
\( x = \beta - 2\alpha + 7 \)
\[ \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 7 + \beta - 2\alpha \\ \beta \\ \alpha \end{pmatrix} = \begin{pmatrix} 7 \\ 0 \\ 0 \end{pmatrix} + \beta \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} + \alpha \begin{pmatrix} -2 \\ 0 \\ 1 \end{pmatrix} \]
Figure 10
A sketch of a three-dimensional coordinate
system with a vertical z-axis arrow and two
other axis arrows. A single tilted plane is
drawn twice, once outlined in red and once
in blue, lying almost exactly on top of each
other to show that the two planes are the
same. From the origin two magenta direction
vectors point outward within the plane,
representing the two free-parameter
direction vectors of the solution. A red
label to the upper right reads 2 planes the
same, with a red arrow pointing to the
overlapping planes.
2 planes the same
Ex8
\[ \begin{cases} x - y + 2z = 7 \\ 3x - 3y + 6z = 2 \end{cases} \] \[ \left(\begin{array}{ccc|c} 1 & -1 & 2 & 7 \\ 3 & -3 & 6 & 2 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & -1 & 2 & 7 \\ 0 & 0 & 0 & -19 \end{array}\right) \]no solution
Figure 11
A three-dimensional sketch showing
coordinate axes (x, y, and z) with two flat
planes drawn parallel to each other. The
upper plane is outlined in red and the lower
plane is outlined in blue. The two planes
are separated by a gap and never meet. A red
arrow points to the lower blue plane with
the label reading 2 unequal, parallel
planes, no intersection, illustrating that
the system has no solution.
2 unequal, parallel planes, no intersection
Ex9
\[ \begin{cases} x + y - z = 2 \\ 3x - y + 2z = 0 \\ 2x - 2y + 3z = -2 \end{cases} \] \[ \left(\begin{array}{ccc|c} 1 & 1 & -1 & 2 \\ 3 & -1 & 2 & 0 \\ 2 & -2 & 3 & -2 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & 1 & -1 & 2 \\ 0 & -4 & 5 & -6 \\ 0 & -4 & 5 & -6 \end{array}\right) \] \[ \rightarrow \left(\begin{array}{ccc|c} 1 & 1 & -1 & 2 \\ 0 & -4 & 5 & -6 \\ 0 & 0 & 0 & 0 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & 1 & -1 & 2 \\ 0 & 1 & -\frac{5}{4} & \frac{3}{2} \\ 0 & 0 & 0 & 0 \end{array}\right) \]\( z = \alpha \) (free)
\( y = \frac{3}{2} + \frac{5}{4}\alpha \)
\( x = 2 - y + z = 2 - \frac{3}{2} - \frac{5}{4}\alpha + \alpha \)
\[ \begin{gathered} = \frac{1}{2} - \frac{1}{4}\alpha \\[0.5em] \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} \frac{1}{2} - \frac{1}{4}\alpha \\ \frac{3}{2} + \frac{5}{4}\alpha \\ \alpha \end{pmatrix} = \begin{pmatrix} \frac{1}{2} \\ \frac{3}{2} \\ 0 \end{pmatrix} + \alpha \begin{pmatrix} -\frac{1}{4} \\ \frac{5}{4} \\ 1 \end{pmatrix} \end{gathered} \]
Figure 12
A three-dimensional sketch on x-y-z
coordinate axes drawn in black, with the
vertical axis pointing up, one axis pointing
to the right, and one axis pointing toward
the lower left. Three planes are drawn in
blue, each shown as a tilted quadrilateral,
with dashed blue lines indicating hidden
edges. All three planes pass through a
single common straight line drawn in red
running roughly vertically through the
origin region. A red arrow points up toward
this line of intersection. Handwritten red
text below reads: 3 planes intersect at a
line.
3 planes intersect at a line.
Ex10
\[ \begin{cases} x + y - z = 2 \\ 2x - y + 3z = 1 \\ x - 2y + 4z = 0 \end{cases} \] \[ \left(\begin{array}{ccc|c} 1 & 1 & -1 & 2 \\ 2 & -1 & 3 & 1 \\ 1 & -2 & 4 & 0 \end{array}\right) \rightarrow \left(\begin{array}{ccc|c} 1 & 1 & -1 & 2 \\ 0 & -3 & 5 & -3 \\ 0 & -3 & 5 & -2 \end{array}\right) \] \[ \rightarrow \left(\begin{array}{ccc|c} 1 & 1 & -1 & 2 \\ 0 & -3 & 5 & -3 \\ 0 & 0 & 0 & 1 \end{array}\right) \]← no solutions.
Figure 13
A three-dimensional sketch on black x-y-z
coordinate axes. Three blue planes are drawn
cutting through space. Their pairwise
intersections form three thick blue parallel
line segments running in the same direction,
none of which meet at a common point. Red
curved arrows point to these segments and
are labeled '3 parallel lines'. A caption
below states the three planes form three
parallel lines with no common intersection.
3 parallel lines
3 planes forming 3 parallel lines with no common intersection.