What value of m makes the following linear system of equations have infinitely many situations.

-6x+5y=14
12x+my=-15

Also, does the value of 3 for m give this system of equations one solution. Lastly, does the value of -10 in this system of equations give no solution?

If you can also provide me with the steps for what value of m would give you infinite solutions, that would be great, given that I would like to be able to solve these type of problems on my own.

for infinitely many solutions, the two equations must describe the same line

-6x+5y=14
multiply everything by -2 and you have
12x-10y = -28

The line 12x-10y = -15 provides no solutions, since the two lines are parallel, and do not intersect

setting m=3 makes the two slopes different, so the lines must intersect.

To determine the value of "m" that makes the linear system have infinitely many solutions, we need to find the condition under which the system of equations becomes dependent. In other words, the determinant of the coefficient matrix must be zero.

The given system of equations is:
-6x + 5y = 14 ...(1)
12x + my = -15 ...(2)

First, let's express equation (2) in standard form (ax + by = c):
my + 12x = -15 ...(3)

To find the determinant of the coefficient matrix, let's set up the matrix as follows:
| -6 5 |
| 12 m |

The determinant is calculated as (-6 * m) - (5 * 12) = -6m - 60.

For the system to have infinitely many solutions, the determinant must be zero:
-6m - 60 = 0

Solving this equation:
-6m = 60
m = 60 / -6
m = -10

Therefore, the value of m that makes the system have infinitely many solutions is m = -10.

Now let's check if m = 3 and m = -10 give one solution or no solution.

For m = 3:
-6x + 5y = 14
12x + 3y = -15

The system has a unique solution since the determinant of the coefficient matrix (3 * -6 - 5 * 12) is not zero.

For m = -10:
-6x + 5y = 14
12x - 10y = -15

The system does not have a solution since the determinant of the coefficient matrix (-10 * -6 - 5 * 12) is zero.

In conclusion:
- The value of 3 for m gives the system of equations one solution.
- The value of -10 for m gives the system of equations no solution.