Lesson Geometry interpretation of the linear system of two equations with two unknowns

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<H2>Geometry interpretation of the linear system of two equations with two unknowns</H2>

Geometry interpretation of a linear system of two equations with two unknowns is the geometry presentation of 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 lessons 
<A HREF= http://www.algebra.com/algebra/homework/coordinate/lessons/Solution-of-the-lin-system-of-two-eqns-by-the-Subst-method.lesson> Solution of the linear system of two equations with two unknowns by the Substitution method</A> and 
<A HREF= http://www.algebra.com/algebra/homework/coordinate/lessons/Solution-of-the-lin-syst-of-two-eqns-with-two-unknowns-Elimination-method.lesson> Solution of the linear system of two equations with two unknowns by the Elimination method</A> in this module.
This lesson is focused on the geometry interpretation and its goal of  to extend your ability to analyze such systems of equations.

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<H3>Example 1</H3>

Let us consider the system of equations

{{{system (2x +  y =  5,
          -4x + 6y = -2
)}}}

<B>Figure 1</B> to the right shows two straight lines {{{2x+y=5}}} (in black) and {{{-4x+6y=-2}}} (in blue).
These straight lines intersect at the point <B>(x,y) = (2,1)</B>. 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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
equations is <B>consistent</B> and equations are <B>independent</B>.
 </TD>
  <TD>
{{{drawing( 200, 200, -2, 4, -2, 4, 
            grid(1),
            line   (-2,   9,   4, -3),
            locate ( 2.2,-0.6, 2x+y=5),

blue(line   (-2, -1.666,   4, 2.333)),
            locate (1.9, 2.7,    -4x+6y=-2)
)}}}
<B>Figure 1. Geometry presentation</B> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>to Example 1</B>
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<H3>Example 2</H3>

Let us consider the system of equations

{{{system (3x - 2y =  5,
           5x -  y =  6
)}}}

<B>Figure 2</B> 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 <B>(x,y) = (1,-1)</B>, which 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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
equations is <B>consistent</B> and equations are <B>independent</B>.
 </TD>
  <TD>
{{{drawing( 200, 200, -2, 4, -3, 3, 
            grid(1),
            line   (-2,  -5.5,   4, 3.5),
            locate ( 2.2, 0.9, 3x-2y=5),

blue(line   (-2, -16,     4, 14)),
            locate (0.9, 2.7,     5x-y=6)
)}}}
<B>Figure 2. Geometry presentation</B> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>to Example 2</B>
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<H3>Example 3</H3>

Let us consider the system of equations

{{{system (3x - 2y =  5,
          12x - 8y = 20
)}}}

<B>Figure 3</B> 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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
equations is <B>consistent</B> and equations are <B>dependent</B>.
 </TD>
  <TD>
{{{drawing( 200, 200, -2, 4, -3, 3, 
            grid(1),
            line   (-2,  -5.5,   4, 3.5),
            locate ( 1.5, 0.9,   3x-2y=5),

blue(line   (-2,  -5.5,   4, 3.5)),
            locate (1.4,  2.7,  12x-8y=20)
)}}}
<B>Figure 3. Geometry presentation</B> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>to Example 3</B>
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<H3>Example 4</H3>

Let us consider the system of equations

{{{system (3x - 2y =  5,
          12x - 8y = 15
)}}}

<B>Figure 4</B> 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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
equations is <B>inconsistent</B> and equations are <B>dependent</B>.
 </TD>
  <TD>
{{{drawing( 200, 200, -2, 4, -3, 3, 
            grid(1),
            line   (-2,  -5.5,   4, 3.5),
            locate ( 2.1, 0.7,   3x-2y=5),

blue(line   (-2,  -4.875, 4, 4.125)),
            locate (0.4,  2.7,   12x-8y=15)
)}}}
<B>Figure 4. Geometry presentation</B> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>to Example 4</B>
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</TABLE> 
The Table below lists, in the ordered form, properties of a linear system of two equations with two unknowns along with the corresponding properties of their geometry 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.
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&nbsp;&nbsp;
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<B>The solution existence and multiplicity</B>
 </TD>
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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Terminology</B>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;<B>The coefficient properties</B>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
  <TD>
&nbsp;&nbsp;&nbsp;&nbsp;<B>Geometry presentation</B>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE> 
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1.
&nbsp;&nbsp;
&nbsp;&nbsp;
 </TD>
  <TD>
The system of equations has a solution,
and the existing solution is unique.
&nbsp;&nbsp;
 </TD>
  <TD>
The system of equations is <B>consistent</B>,
and equations are <B>independent</B>.
&nbsp;&nbsp;
 </TD>
  <TD>
The row of coefficients 
of the first equation is not proportional 
to that of the second equation.
 </TD>
  <TD>
In the geometry presentation 
two straight lines intersect 
and have only one common point.
 </TD>
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2.
&nbsp;&nbsp;
&nbsp;&nbsp;
&nbsp;&nbsp;
 </TD>
  <TD>
The system of equations has more&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
than one solution.
(==) The system of equations has 
infinitely many solutions.
 </TD>
  <TD>
The system of equations is <B>consistent</B>,
and equations are <B>dependent</B>.
&nbsp;&nbsp;
&nbsp;&nbsp;
 </TD>
  <TD>
The row of coefficients and the right&nbsp;&nbsp;&nbsp;&nbsp; 
side of the first equation is 
proportional to that of the second 
equation.
 </TD>
  <TD>
In the geometry presentation 
two straight lines coincide 
and have all points common.
&nbsp;&nbsp;
 </TD>
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3.
&nbsp;&nbsp;
&nbsp;&nbsp;
&nbsp;&nbsp;
 </TD>
  <TD>
The system of equations has no&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
solutions.
&nbsp;&nbsp;
&nbsp;&nbsp; 
 </TD>
  <TD>
The system of equations is <B>inconsistent</B>,
and equations are <B>dependent</B>.
&nbsp;&nbsp;
&nbsp;&nbsp;
 </TD>
  <TD>
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.
 </TD>
  <TD>
In the geometry presentation 
two straight lines are parallel 
and have no common points.
&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE>