Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets.

4x - 3y = 6
-12x + 9y = -24

Question

Solve the system by the method of your choice. Identify systems with no solution and systems with infinitely many solutions, using set notation to express their solution sets. 

4x - 3y = 6 
-12x + 9y = -24

mathstudent55 · Answer

Look at both equations.
Before you start solving the system, is there a common factor in all terms of the second equation?
Is there a common factor in all terms of the second equation?

nthenic_oftime · Answer

yes @mathstudent55 im sorry i got busy for a moment. first the common factpor of 2 and 3 for the second equation...

mathstudent55 · Answer

The first equation has coefficients 4, -3, and 6. It has no common factors.
The second equation has coefficients -12, 9, and -24.
-12, 9, and -24 are all divisible by 3.
3 is the common factor for the second equation.

We divide every term of the second equation by 3:

-4x + 3y = -8

Now we compare it with the first equation.

4x - 3y = 6

Let's multiply the second equation by -1 and write it below the first equation:

4x - 3y = 6
4x - 3y = 8

As you can see, the equations are very similar. The only difference between them is the 6 and the 8.

mathstudent55 · Answer

Now let's try to solve it by the addition method.
Multiply the second equation by -1, and write it below the first equation. 
Then add the equations.

  4x - 3y =   6 
-4x + 3y = -8
---------------
  0 + 0     = -2

When we add the equations, we get this statement.
0 = -2

Since 0 = -2 is a false statement, that means there is no solution for this system of equations. The solution set is the empty set.