Question

Solve the following system of equations:
x - 2y = 6
2x - 4y = 10

Infinitely many solutions

No solutions

(0, 0)

(6, 10)

No solutions right?

Answer 1

Think about it like this. These are equations for two lines, both of which have slopes and cross certain points. If they are multiples of each other, then they lie right atop one another. If their expressions (the part to the left of the equal sign here) are multiples, but not the answer, then they have the same slope, but cross different points. That means they would be parallel, so there would be no solutions (since a solution is simply a point where both lines cross). Any other cases would just be two lines with nothing in common (different slopes) and only cross at one point. Which do you think this example is, and why?

Answer 2

I think it's infinitely many or none because

x-2y=6
2x-4y=10

2(x-2y=6)

2x-4y=12
2x-4y=10

0=2

Answer 3

Forgot to mention, if they lie on top of each other, that means there are infinitely many solutions.

Answer 4

so even though it says 0=2 there are infinitely many?

Answer 5

And yes you are correct. So what can you infer from the math you just did?

Answer 6

No, I was just clarifying my first statement.

Answer 7

I can infer there are no solutions

Answer 8

There you go, because the two lines are parallel. It wasn't completely necessary to see that relation, but I feel it makes things easier in the long run.