Lesson Geometric interpretation of a linear system of two equations in two unknowns

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Geometric interpretation of a linear system of two equations in two unknowns

Geometric interpretation of a linear system of two equations in two unknowns is the geometric presentation of the two straight lines that correspond to linear functions
of these equations. It gives you a visual image showing immediately if the given linear system is consistent or inconsistent; whether the equations are dependent
or not. Below are some examples.
Note that the systems of equations in these examples were just considered from the point of view of their solutions in the lessons
- Solution of the linear system of two equations with two unknowns by the Substitution method and
- Solution of the linear system of two equations with two unknowns by the Elimination method in this module.
This lesson is focused on the geometric interpretation, and its goal is to extend your ability to analyze such systems of equations.

Example 1

Let us consider a system of equations

system+%282x+%2B++y+=++5%2C%0D%0A++++++++++-4x+%2B+6y+=+-2%0D%0A%29

Figure 1 to the right shows two straight lines 2x%2By=5

(in black) and -4x%2B6y=-2

(in blue).
These straight lines intersect at the point (x,y) = (2,1). Note that this intersection point is exactly the solution
of the system (see the lessons referred above).
The fact that these straight lines intersect and have only one common point corresponds to that the system of
equations is consistent and the equations are independent.

Figure 1. Geometric presentation
to Example 1

Example 2

Let us consider a system of equations

system+%283x+-+2y+=++5%2C%0D%0A+++++++++++5x+-++y+=++6%0D%0A%29

Figure 2 to the right shows two straight lines 3x-2y=5

(in black) and 5x-y=6

(in blue).
These straight lines intersect, and the intersection point is (x,y) = (1,-1). It is exactly the solution of the system
(see the lessons referred above).
The fact that these straight lines intersect and have only one common point corresponds to that the system of
equations is consistent and the equations are independent.

Figure 2. Geometric presentation
to Example 2

Example 3

Let us consider a system of equations

system+%283x+-+2y+=++5%2C%0D%0A++++++++++12x+-+8y+=+20%0D%0A%29

Figure 3 to the right shows two straight lines 3x-2y=5

(in black) and 12x-8y=20

(in blue).
These straight lines actually coincide, so all their points are common and all represent solutions of the equation system
(see the lessons referred above).
The fact that these straight lines coincide and have all points common corresponds to that the system of
equations is consistent and the equations are dependent.

Figure 3. Geometric presentation
to Example 3

Example 4

Let us consider a system of equations

system+%283x+-+2y+=++5%2C%0D%0A++++++++++12x+-+8y+=+15%0D%0A%29

Figure 4 to the right shows two straight lines 3x-2y=5

(in black) and 12x-8y=15

(in blue).
These straight lines are actually parallel, so the lines have no common points
(see the lessons referred above).
The fact that these straight lines are parallel and have no common points corresponds to that the system of
equations is inconsistent and the equations are dependent.

Figure 4. Geometric presentation
to Example 4

The Table below lists, in the ordered form, properties of a linear system of two equations in two unknowns along with the corresponding properties of their
geometric presentations. Equivalent properties are listed in the row cells of the Table, from the left to the right, while the different properties are listed
in the column cells, from the top to the bottom.

The solution existence and multiplicity

Terminology

The coefficient properties

Geometric presentation

The system of equations has a solution,
and the existing solution is unique.

The system of equations is consistent,
and equations are independent.

The row of coefficients
of the first equation is not proportional
to that of the second equation.

In the geometric presentation
two straight lines intersect
and have only one common point.

The system of equations has more
than one solution.
(==) The system of equations has
infinitely many solutions.

The system of equations is consistent,
and equations are dependent.

The row of coefficients and the right
side of the first equation is
proportional to that of the second
equation.

In the geometric presentation
two straight lines coincide
and have all points common.

The system of equations has no
solutions.

The system of equations is inconsistent,
and equations are dependent.

The row of coefficients of the first
equation is proportional to that of the
second equation, but the right sides
are not proportional in this way.

In the geometric presentation
two straight lines are parallel
and have no common points.

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