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

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<H2>Geometric interpretation of a linear system of two equations in two unknowns</H2>

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. &nbsp;It gives you a visual image showing immediately if the given linear system is consistent or inconsistent; &nbsp;whether the equations are dependent 
or not. &nbsp;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

&nbsp;&nbsp;&nbsp;&nbsp;- <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> &nbsp;and

&nbsp;&nbsp;&nbsp;&nbsp;- <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>&nbsp; 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.

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<H3>Example 1</H3>

Let us consider a system of equations

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

<B>Figure 1</B>&nbsp; to the right shows two straight lines &nbsp;{{{2x+y=5}}}&nbsp; (in black) and &nbsp;{{{-4x+6y=-2}}}&nbsp; (in blue).
These straight lines intersect at the point &nbsp;<B>(x,y) = (2,1)</B>. &nbsp;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 &nbsp;<B>consistent</B>&nbsp; and the equations are &nbsp;<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. Geometric presentation</B> 
&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 a system of equations

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

<B>Figure 2</B>&nbsp; to the right shows two straight lines &nbsp;{{{3x-2y=5}}}&nbsp; (in black) &nbsp;and &nbsp;{{{5x-y=6}}}&nbsp; (in blue).
These straight lines intersect, &nbsp;and the intersection point is &nbsp;<B>(x,y) = (1,-1)</B>. &nbsp;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&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
equations is &nbsp;<B>consistent</B>&nbsp; and the equations are &nbsp;<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. Geometric presentation</B> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>to Example 2</B>
  </TD>
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  <TD>
<H3>Example 3</H3>

Let us consider a system of equations

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

<B>Figure 3</B>&nbsp; to the right shows two straight lines &nbsp;{{{3x-2y=5}}}&nbsp; (in black) &nbsp;and &nbsp;{{{12x-8y=20}}}&nbsp; (in blue).
These straight lines actually coincide, &nbsp;so all their points are common and all represent solutions of the equation system&nbsp;&nbsp;&nbsp;&nbsp;
(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 &nbsp;<B>consistent</B>&nbsp; and the equations are &nbsp;<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. Geometric presentation</B> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>to Example 3</B>
  </TD>
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  <TD>
<H3>Example 4</H3>

Let us consider a system of equations

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

<B>Figure 4</B>&nbsp; to the right shows two straight lines &nbsp;{{{3x-2y=5}}}&nbsp; (in black) &nbsp;and &nbsp;{{{12x-8y=15}}}&nbsp; (in blue).
These straight lines are actually parallel, &nbsp;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;&nbsp;
equations is &nbsp;<B>inconsistent</B>&nbsp; and the equations are &nbsp;<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. Geometric presentation</B> 
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>to Example 4</B>
  </TD>
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</TABLE> 
The Table below lists, &nbsp;in the ordered form, &nbsp;properties of a linear system of two equations in two unknowns along with the corresponding properties of their

geometric presentations. &nbsp;Equivalent properties are listed in the row cells of the Table, &nbsp;from the left to the right, &nbsp;while the different properties are listed 
in the column cells, &nbsp;from the top to the bottom.
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&nbsp;&nbsp;
 </TD>
  <TD>
<B>The solution existence and multiplicity</B>
 </TD>
  <TD>
&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;&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;&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
  <TD>
&nbsp;&nbsp;<B>Geometric presentation</B>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE> 
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  <TD>
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 geometric presentation 
two straight lines intersect 
and have only one common point.
 </TD>
 </TR>
</TABLE> 
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  <TD>
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 geometric presentation 
two straight lines coincide 
and have all points common.
&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE> 
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  <TR>
  <TD>
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 geometric presentation 
two straight lines are parallel 
and have no common points.
&nbsp;&nbsp;
 </TD>
 </TR>
</TABLE>

My other lessons on solution of the linear system of two equations in two unknowns in this site are

&nbsp;&nbsp;&nbsp;&nbsp;- <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 a linear system of two equations in two unknowns by the Substitution method</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <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 a linear system of two equations in two unknowns by the Elimination method</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/coordinate/lessons/Solution-of-the-lin-syst-of-two-eqns-with-two-unknowns-using-det.lesson>Solution of a linear system of two equations in two unknowns using determinant</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Useful-tricks-when-solving-systems-of-2-eqns-in-2-unknowns-by-the-Subst-method.lesson>Useful tricks when solving systems of 2 equations in 2 unknowns by the Substitution method</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =http://www.algebra.com/algebra/homework/coordinate/lessons/Solving-word-probs-using-linear-systems-of-two-eqns-with-two-unknowns.lesson>Solving word problems using linear systems of two equations in two unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Word-problems-that-lead-to-a-simple-system-of-two-equations-in--wo-unknowns.lesson>Word problems that lead to a simple system of two equations in two unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/Oranges-and-Grapefruits.lesson >Oranges and grapefruits</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Using-systems-of-equations-to-solve-problems-on-Tickets.lesson>Using systems of equations to solve problems on tickets</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Three-methods-for-solving-standard-typical-problem-on-tickets.lesson>Three methods for solving standard (typical) problems on tickets</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Using-systems-of-equations-to-solve-problems-on-SHARES.lesson>Using systems of equations to solve problems on shares</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Using-systems-of-equations-to-solve-problems-on-investment.lesson>Using systems of equations to solve problems on investment</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Two-mechanics-work-on-a-car.lesson>Two mechanics work on a car</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/The-Robinson-family-and-the-Sanders-family-each-used-their-sprinklers-last-summer.lesson>The Robinson family and the Sanders family each used their sprinklers last summer</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Roses-and-violets.lesson>Roses and violets</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/Counting-calories-and-grams-of-fat-in-combined-food.lesson>Counting calories and grams of fat in combined food</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/A-theater-group-made-appearances-in-two-cities.lesson>A theater group made appearances in two cities</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Exchange-problems-solved-using-systems-of-linear-equations.lesson>Exchange problems solved using systems of linear equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Typical-word-problems-on-systems-of-2-equations-in-2-unknowns.lesson>Typical word problems on systems of 2 equations in 2 unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/How-to-algebreze-and-to-solve-this-problem-on-2-eqns-in-2-unknwns.lesson>HOW TO algebraize and solve these problems on 2 equations in 2 unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/One-unusual-problem-to-solve-using-system-of-two-equations.lesson>One unusual problem to solve using system of two equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Non-standard-problem-with-a-tricky-setup-.lesson>Non-standard problems with a tricky setup</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/Sometime-one-equations-is-enough-to-find-two-unknowns-by-an-UNIQUE-way.lesson>Sometimes one equation is enough to find two unknowns in a unique way</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Solving-mentally-word-problems-on-two-equations-in-two-unknowns-NEW.lesson>Solving mentally word problems on two equations in two unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Word-problem-to-solve-combining-system-of-linear-equations-and-a-price-equation.lesson>Word problem to solve combined system of linear equations and a price equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Solving-systems-of-non-linear-equations-by-reducing-to-linear-ones.lesson>Solving systems of non-linear equations by reducing to linear ones</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/Solving-word-problems-for-3-unknowns-by-reducing-to-equations-in-2-unknowns.lesson>Solving word problems for 3 unknowns by reducing to equations in 2 unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/coordinate/lessons/System-of-equations-helps-to-solve-a-problem-for-the-Thanksgiving-day.lesson>System of equations helps to solve a problem for the Thanksgiving day</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF =https://www.algebra.com/algebra/homework/coordinate/OVERVIEW-of-lessons-on-solving-systems-of-two-linear-equations-in-two-unknowns.lesson>OVERVIEW of lessons on solving systems of two linear equations in two unknowns</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.