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Infinitely Many Solutions or No Solution? Equations Special Cases

take this first example here so the

first thing we do when we solve

equations is we try to simplify the left

side and simplify the right side that's

the first step so if I do that I'm going

to use the distributive property to get

3x minus 6 now when you look here you'll

say wait a second the left side and the

right side they're identical right

they're exactly the same so they call

this an identity what that means is you

know okay the left on the right sides

are exactly the same they're identical

like identical twins whatever you put in

for X the left side and the right side

will always be equal so no matter what

you put in for X it's going to be true

so this means this has an infinite

number of solutions so you could say

infinitely many you could say you know

infinite number of solutions any one of

those would work okay or you could say

it's an identity now this one was kind

of obvious but let's say you didn't

notice that first and just we're going

through the motions and you were just

solving the equation so maybe you

subtracted 3x from both sides and then

you said okay these cancel oh wait a

second what happened my variable just

kind of disappeared right and so you're

left with negative 6 equals negative 6

you say wait a second what do I do from

here well one thing you notice is that

this is always true negative 6 will

always equal negative 6 it doesn't

matter that variable doesn't even come

into play here it cancelled out on both

sides so you can tell it from this case

hmm all real solutions all real numbers

infinitely many solutions however you

want to say it that would be the answer

to this problem

okay now an example number B letter B I

should say same process you want to

simplify the left side simplify the

right side as much as you can so I'm

just going to combine like terms that's

a negative 3x plus 4 equals a negative

3x plus 10 ok now if I add 3x to both

sides okay look what happens these

cancel out I get 4 equals 10 now if I

think about that for a minute I say mmm

wait a second 4 will never equal 10 that

doesn't even make any sense this is a

nonsensical you know answer it doesn't

make any sense

what this means is that no matter what

you put in for X because these

quantities are the same on both sides

the 3x the negative 3x well that tells

you is that then they're gonna cancel

out regardless of what X is and that 4

will never equal 10 so it doesn't matter

what you put in for X this will never be

true so this means that this is no

solution okay no solution or no answer

or not

possible or you know empty set any one

of these would would work or depending

on what your teacher prefers so you can

see we either get an identity they're

identical infinite solutions or we get

something that doesn't make any sense

and that's no solution so I hope this

helped you understand how to work with

some special cases when you're solving

equations subscribe to the channel check

out some of my other videos on Mario's

math tutor and YouTube channel and I

look forward to helping you in the

future videos I'll talk to you soon