Question 743668


If these two or more linear equations {{{intersect}}}, that point of intersection is called the {{{solution}}} to the system of linear equations.

How many solutions can systems of linear equations have?

Case I: {{{1}}} solution

This is the most common situation and it involves lines that intersect {{{exactly}}} {{{1}}} time. 

Case 2: {{{No}}} {{{solutions}}}

This only happens when the lines are {{{parallel}}}. As you can see, parallel lines are {{{not}}} going to {{{ever}}}{{{ meet}}}. 

Case 3: {{{Infinite}}} {{{solutions}}}

This is the {{{rarest}}} case and only occurs when you have the {{{same}}}{{{ line}}}.

Consider, for instance, the two lines below 

{{{y = 2x+1}}} and 

{{{2y = 4x +2}}} 

These two equations are really the {{{same}}} line.
 
Example of a system that has infinite solutions:

    Line 1: {{{y = 2x + 1}}}

find {{{x}}} and {{{y-intercept}}}

set {{{x=0}}} and find {{{y-intercept}}}

{{{y = 2*0 + 1}}} => {{{y = 1}}}; so, {{{y-intercept}}} is at ({{{0}}},{{{1}}})

set {{{y=0}}} and find {{{x-intercept}}}

{{{0 = 2x + 1}}} => {{{2x = -1}}}=> {{{x = -1/2}}}; so, {{{x-intercept}}} is at ({{{-1/2}}},{{{0}}})


    Line 2: {{{2y = 4x + 2}}}

set {{{x=0}}} and find {{{y-intercept}}}

{{{2y = 4*0 + 2}}} => {{{2y =2}}}=> {{{y =1}}}; so, {{{y-intercept}}} is at ({{{0}}},{{{1}}})

set {{{y=0}}} and find {{{x-intercept}}}

{{{2*0= 4x + 2}}} => {{{0-2 =4x}}}=> {{{x = -2/4}}}=> {{{x = -1/2}}}; so, {{{x-intercept}}} is at ({{{-1/2}}},{{{0}}})


on a graph you will see only one line:


{{{ graph( 600, 600, -10,10, -10, 10, 2x + 1, 2x + 1) }}}