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One Solution, No Solution, or Infinitely Many Solutions - Consistent & Inconsistent Systems

given a system of two equations how can

you tell

if it's consistent or inconsistent if

it's dependent or independent let's go

over the important things they need to

know let's just keep it simple

so if you have a system of equation that

has only one solution then it's going to

be consistent and it's going to be

independent if there's only one solution

now sometimes you might have many

solutions as opposed to one solution so

if you have many solutions will it be

consistent or inconsistent it turns out

that it's still going to be consistent

do you think is going to be dependent or

independent if there's many solutions

it's dependent as opposed to independent

now sometimes there's going to be no

solution if there's no solution it's

going to be inconsistent

in addition it's going to be independent

if there's no solution so make sure you

know this information and that we'll go

through a few examples using this

information so how can we distinguish

between one solution no solution and

many solutions let's say if you solve a

system of equations and you get one

value for X and one value for one it's

going to be one solution

now what about no solution when you're

solving it if you get to a point let's

say like two equals five that is not a

true statement so in a situation like

this it's a null solution

now what about many solutions how does

that look like whenever you solve a

system of two equations if you get

something that looks like this is 0

equals 0 5 equals 5 or x equals x if the

two sides are exactly the same

then it's many solutions but if you get

x equals a number

rather than itself it's a one solution

so that's how you can distinguish

between these three categories and once

you know it's a one solution

you know it's consistent and independent

if it's many solutions you know it's

consistent and dependent if it's a no

solution it's inconsistent and now let's

start with this example 3x plus y is

equal to 17 and also 4x minus y is equal

to 18 determine if there's one solution

and no solution or many solutions if

it's consistent or inconsistent

dependent or independent well let's use

the elimination method to get the answer

if we add the two equations y and

negative Y will cancel 3x plus 4x is 7 X

17 plus 18 is 35

now let's divide both sides by 7 35

divided by 7 is 5 now that we have the x

value let's plug it into the first

equation to get the Y value so 3 times 5

plus y is equal to 17 3 times 5 is 15

and 17 minus 15 is 2 so y is equal to 2

so there's only one solution it's 5

comma 2 and whenever there's one

solution is it going to be consistent or

inconsistent one solution will always be

associated with consistent and anytime

you have one solution it's going to be

independent it's always gonna work out

that way

here's the next example 2x plus 4y is

equal to 8 and also X plus 2y is equal

to 4 determine if this system of

equations if it's consistent or

inconsistent dependent or independent if

it contains one solution no solution or

many solutions well let's use

elimination again let's multiply the

second equation by negative 2 so first

let's you write the first equation which

is 2x plus 4y is equal to 8 now for the

second equation x times negative 2

that's going to be negative 2x 2y times

a negative 2 is negative 4 1 4 times

negative 2 is negative 8 if we add the 2

equations negative 2x plus 2x is 0 for y

plus a negative 4 y is 0 8 minus 8 is 0

so 0 equals 0 this is the case where we

have many solutions now if there are

many solutions then it's going to be

consistent

but dependent and so that's it for this

problem try this one

3x plus 2y is equal to 5 and 6x plus 4y

is equal to 8 is there going to be one

solution no solution or many solutions

well let's multiply the first equation

by negative 2 3x times negative 2 is

negative 6 X 2y times negative 2 is

negative 4y + 5 times negative 2 is

negative 10 and let's rewrite the second

equation so now if we add it negative 6x

plus 6x it cancels a negative 4y plus 4y

cancels so that's simply 0 negative 10

plus 8 is negative 2

now 0 does not equal negative 2 so this

is the case where we have no solution

and when there's no solution it is

inconsistent it's inconsistent but it's

independent as well and so that's it for

this example