We will solve this system with method of opposing coefficients - Gaussian algorithm
-2x+6y= -38
3x-4y= 32
We will divide first equation with number 2 and get
-x+3y = - 19
then we will multiply the same equation with number 3 and get
-3x+9y = - 57
We will overwrite the second equation below the last one and get next equivalent system
-3x+9y = - 57
3x-4y = 32
We add first equation to the second and get
5y = -25 => y= -25/5 => y= -5
Now we will replace variable y= -5 in the equation -x+3y = - 19 and get
-x+3(-5) = -19 => -x-15 = -19 => x- 19-15=4 => x=4
The correct answer is (x,y) = (4,-5)
We can check in the first equation and get
-2*4+6*(-5) = -38
-8-30 = -38
-38 = -38 We get equality, the solutions are correct.
This system have one real solution.
Good luck!!!