Question

How many solutions are there to the following system of equations? 3x-9y=0 -x+3y=-3 A. 0 B. 1 C. Infinitely many

Answer 1

A signle system of linear equations has infinitely many solutions.
Two identical systems of linear equations have infinitely many solutions.
Two systems of linear equations has just one solution.
Two systems of parallel linear equations will has no solutions.

We have two systems of linear equations.

Let's see if they are parallel.
Parallel lines have the same slope.
We'll see if they do by converting to slope-intercept form.
(this will also let us know if they are identical)

1st Eqn.
3x - 9y = 0
Keep y on the left, x and the constant on the right...
-9y = -3x + 0
Divide by the y coefficient.
y = 1/3x + 0

2nd Eqn.
-x + 3y = -3
Keep y on the left, x and the constant on the right...
3y = x - 3
Divide by the y coefficient.
y = 1/3x - 3

As you can see, the two equations have the same slope! (1/3)
Because of this, they are parallel.
And they are not identical because the y-intercepts are different.

There are no solutions to this system of equations.

Answer 2

Equation 1: 3x-9y=0
Equation 2: -x+3y=-3

Multiply Equation 2 x 3:
-3x+9y=-9
Therefore,

3x-9y=0
-3x+9y=-9
~~~~~~~
0=-9
The answer is A. 0, because 0 doesnt equal -9