solve by elimination: -4x-6y=5 and -8x-12y=10. Are they consistent or inconsistent, dependent or independent?

Answer 1

OK

- 4x - 6y = 5
- 8x -12y = 10

Multiply the top equation by (-2)

8x + 12y = -10
- 8x -12y = 10

Now take the sum of the both equations

8x - 8x + 12y - 12y = -10 + 10

0 = 0

This identity is obviously valid, but doesn't give us a condition for either x or y, which means that any x and any y will solve the system.
There are infinitely many solutions.

(Graphically this means that the lines -4x-6y=5 and -8x-12y=10 coincide, i.e. they are one and the same line and as such they have infinitely many points in common )

The system is CONSISTENT (since there is a solution [ie all pairs of real numbers]. But as there are infinitely many solutions (and not only one), the system is DEPENDENT.

Hope this helps!

Answer 2

-4x - 6y = 5
-8x - 12y = 10
Eliminate x:
Multiply 1st equation by 2, and subtract second:
0 = 0.
You have eliminated y as well.
The equations are therefore dependent. The second is the first multiplied by 2.

Answer 3

These equations are dependent.
If we multiply the first by 2 it becomes the second one.
Subtracting gives 0=0, i.e., the equations are
dependent(or redundant).

Answer 4

Unfortunately you can't solve for x and y, since the second equation is the exact same as the first (every term is just doubled).