to solve by determinants, the general concept is:
first create a matrix with the coefficients and the constants of the system of equations.
the system of equations given is:
2x + y + 3z = -9
3x - 2y + 5z = -8
x + 7y - 4z = -1
your matrix will look like this:
2 1 3 -9
3 -2 5 -8
1 7 -4 -1
the first column corresponds to the coefficients of the x variable.
the second column corresponds to the coefficients of the y variable.
the third column corresponds to the coefficients of the z variable.
the fourth column corresponds to the constants.
you first find the determinants of the 3 by 3 matrix formed from the first 3 columns.
that matrix will look like this:
2 1 3
3 -2 5
1 7 -4
the first column corresponds to the coefficients of the x variable.
the second column corresponds to the coefficients of the y variable.
the third column corresponds to the coefficients of the z variable.
that determinant will be equal to 32, which we'll call D.
you then replace the first column with the fourth column and find the determinant of that 3 by 3 matrix.
that matrix will look like this:
-9 1 3
-8 -2 5
-1 7 -4
the first column corresponds to the constants.
the second column corresponds to the coefficients of the y variable.
the third column corresponds to the coefficients of the z variable.
that determinant will be equal to 32, which we'll call Dx.
you then go back to the original matrix and replace the second column with the fourth column and find the determinant of of that 3 by 3 matrix.
that matrix will look like this.
2 -9 3
3 -8 5
1 -1 -4
the first column corresponds to the coefficients of the x variable.
the second column corresponds to the constants.
the third column corresponds to the coefficients of the z variable.
that determinant will be equal to -64, which we'll call Dy.
you then go back to the original matrix and replace the third column with the fourth column and find the determinant of that 3 by 3 matrix.
that matrix will look like this.
x y c
2 1 -9
3 -2 -8
1 7 -1
the first column corresponds to the coefficients of the x variable.
the second column corresponds to the coefficients of the y variable.
the third column corresponds to the constants.
that determinant will be equal to -96, which we'll call Dz.
the value of x is equal to Dx / D which is equal to 32/32 = 1.
the value of y is equal to Dy / D whcih is equal to -64/32 = -2.
the value of z is equal to Dz / d which is equal to -96/32 = -3.
the solution to your problem is:
x = 1
y = -2
z = -3
i solve this problem using online calculators and got the same results so i'm reasonably confident the answer is correct.
here's a reference on how to calculate a determinant.
https://www.mathsisfun.com/algebra/matrix-determinant.html
here's a reference on how to use determinants to solve a system of linear equations.
https://www.youtube.com/watch?v=TtxVGMWXMSE