Question

how many solutions are there to the following system of equations? 3x-9y=0 and -x+3y=-3

Answer 1

Well I would find the number by attempting to solve the system.

Answer 2

You easily solve 3x-9y=0 for x

Answer 3

then you can input that into the other equation -x+3y=-3
and try to solve for y

Answer 4

still confused

Answer 5

can you solve 3x-9y=0 for x?

Answer 6

no i am lost

Answer 7

would you add nine to both sides?

Answer 8

3x-9y=0
Undo the subtraction of 9y by adding 9y (add on both sides)

Answer 9

3x-9y=0
+9y +9y
------------

3x=9y
you would add 9y just like i did right here

Answer 10

now can you get x by itself?

Answer 11

Hint: think dividing something on both sides

Answer 12

divide both sides by 3?

Answer 13

x=9y/3
x=3y since 9/3=3

Answer 14

x=3?

Answer 15

the other equation is -x+3y=-3
right?

Answer 16

so replace the x there with 3y

Answer 17

-(3y)+3y=-3

if you end up with the same thing on both sides you have infinitely many answers
if you end up with something different on both sides you have 0 answers
if you end up with something like y=a number you have one answer

Answer 18

so it will be 0? am i right?

Answer 19

yes because -3y+3y=0
and 0 doesn't equal -3 right?

Answer 20

gj

Answer 21

yeah!!!!!! thank you so much

Answer 22

one more thing:
3x-9y=0
-x+3y=-3
------we can write both in y=mx+b form
3x-9y=0 add 9y on both sides
9y=3x divide by 9 on both sides
y=3/9 x
y=1/3 x

ther other equation is
-x+3y=-3
3y=x-3 I add x on both sides here
y=1/3 x -3/3 I divided 3 on both sides
y=1/3 x - 1 I simplified 3/3=1

now comparing
y=1/3 x
y=1/3 x - 1

They both have the same slope which is 1/3

So if two lines have the same slope, that means 1 of 2 things
1) same line if they have the same y-intercept
2) parallel if they have different y-intercept

The case here is 2) since the first equation has y-intercept 0 and second equation has y-intercept -1

Parallel lines never intersect so that means they will not have s "solution"
The solution is the set in which the lines intersect

Answer 23

That is just another way to look it.
The possible solutions to a system of 2 linear equations with 2 variables is:
1) same slope and same y-intercept => infinitely many solutions (since same line)
2) same slope and different y-intercept=> 0 solutions (since parallel)
3) different slope=> 1 solution