Lesson The difference between no solution and infinite solutions in solving a system of linear equations

Source code of 'The difference between no solution and infinite solutions in solving a system of linear equations'

This Lesson (The difference between no solution and infinite solutions in solving a system of linear equations) was created by by algebrahouse.com(1659)

: View Source, Show
About algebrahouse.com: Visit: algebrahouse.com to ask questions and for notes, examples, and more.

A commonly asked question I often receive on my website, <a href = "http://www.algebrahouse.com/" target = "_blank">www.algebrahouse.com</a>, is identifying the difference between "no solution" and "infinite solution" when solving a system of linear equations.

A solution to a system of linear equations represents where the two lines intersect.

For two lines, there are three different situations that may occur.
1.) The two lines may have 1 point of intersection (one solution)
2.) The two lines may have 0 points of intersection (no solution).  The lines are parallel.
3.) The two lines may have an infinite number of intersecting points (infinite solutions).  The lines are the same.

In identifying the difference between "no solution" and "infinite solutions, the understanding is quite simple:
- When solving, if the variable disappears and you are left with a true statement, such as 3 = 3, then there are infinite solutions.  There are an infinite number of intersecting points, meaning the two lines are the same.
- If the variable disappears and you are left with a false statement, such as 4 = 5, then there is no solution.  There is no point of intersection, meaning the two lines are parallel.

-------------------------------------------------

Solve the system of equations using the substitution method:
2x - y = 8
y = 2x - 3

2x - (2x - 3) = 8   {substituted (2x - 3), in for y, into top equation}
2x - 2x + 3 = 8 {distributed negative sign through parentheses}
3 = 8 {combined like terms}
= no solution

The variable disappeared and you are left with a false statement.  Therefore, there is no solution, and these two lines are parallel.

*[illustration Image_004.png].
 For more help from me, visit: <a href = "http://www.algebrahouse.com/ask-a-question.html" target = "_blank">www.algebrahouse.com</a> 
--------------------------------------------------

Solve the system of equations using the substitution method:
y = 2x - 4
-6x + 3y = -12

-6x + 3(2x - 4) = -12  {substituted (2x - 4), in for y, into bottom equation}
-6x + 6x - 12 = -12 {used distributive property}
-12 = -12 {combined like terms}
= infinite solutions

The variable disappeared and you are left with a true statement.  Therefore, there are an infinite number of intersecting points, and these two lines are the same.

*[illustration Image_009.png]. *[illustration Image_002.png].
 For more help from me, visit: <a href = "http://www.algebrahouse.com/ask-a-question.html" target = "_blank">www.algebrahouse.com</a> 
--------------------------------------------------

Solve the system of equations using the elimination method:
3x + y = -5
6x + 2y = 10

-6x - 2y = 10 {multiplied top equation by -2}
6x + 2y = 10   {bottom equation stays the same}
-------------
0 = 20 {added the two equations}
= no solution

The variable disappeared and you are left with a false statement.  Therefore, there is no solution, and these two lines are parallel.

*[illustration Image_005.png].
 For more help from me, visit: <a href = "http://www.algebrahouse.com/ask-a-question.html" target = "_blank">www.algebrahouse.com</a> 
--------------------------------------------------

Solve the system of equations using the elimination method:
-5x + 4y = 20
10x - 8y = -40

-10x + 8y = 40 {multiplied top equation by 2}
10x - 8y = -40 {bottom equation stays the same}
---------------
0 = 0 {added the two equations}
= infinite solutions

The variable disappeared and you are left with a true statement.  Therefore, there are an infinite number of intersecting points, and these two lines are the same.

*[illustration Image_006.png]. *[illustration Image_007.png].
 For more help from me, visit: <a href = "http://www.algebrahouse.com/ask-a-question.html" target = "_blank">www.algebrahouse.com</a>