Write the equations in matrix form. We will use the descriptors "coefficient matrix", "variable vector", and "constant vector" to describe the parts of this equation from left to right.

Compute the determinant of the coefficient matrix. For this coefficient matrix, the determinant will be

((1*-2*-4)+(1*1*3)+(1*5*1)) - ((1*1*1)+(1*5*-4)+(1*-2*3))

= (8 + 3 + 5) - (1 - 20 - 6) = 16 - (-25) = 41

Create a new matrix (call it "c1") which is the coefficient matrix with the first column replaced by the constant vector:

Find the determinant of this c1 matrix. It will be ...

((10*-2*-4)+(1*1*-1)+(1*3*1)) - ((10*1*1)+(1*3*-4)+(1*-2*-1))

= (80 - 1 + 3) - (10 - 12 + 2) = 82 - 0 = 82

The solution for x1 is the determinant of the c1 matrix divided by the determinant of the coefficient matrix.

x1 = 82/41 = 2

To find x2, we do the same thing, only replacing the second column of the coefficient matrix by the constant vector.

The determinant of c2 is

((1*3*-4)+(10*1*3)+(1*5*-1) - (1*1*-1)+(10*5*-4)+(1*3*3))

= (-12 + 30 - 5) - (-1 - 200 + 9) = 13 + 192 = 205

The solution for x2 is the determinant of the c2 matrix divided by the determinant of the coefficient matrix.

x2 = 205/41 = 5

To find x3, we do the same thing, only replacing the second column of the coefficient matrix by the constant vector.

The determinant of c3 is

((1*-2*-1)+(1*3*3)+(10*5*1)) - ((1*3*1)+(1*5*-1)+(10*-2*3))

= (2 + 9 + 50) - (3 - 5 - 60) = 61 + 62 = 123

The solution for x3 is the determinant of the c3 matrix divided by the determinant of the coefficient matrix.

x3 = 123/41 = 3

Thus, x1=2, x2=5, x3=3.

| 1 1 1| |x1| |10|

| 5 -2 1|*|x2| = | 3|

| 3 1 -4| |x3| |-1|

Compute the determinant of the coefficient matrix. For this coefficient matrix, the determinant will be

((1*-2*-4)+(1*1*3)+(1*5*1)) - ((1*1*1)+(1*5*-4)+(1*-2*3))

= (8 + 3 + 5) - (1 - 20 - 6) = 16 - (-25) = 41

Create a new matrix (call it "c1") which is the coefficient matrix with the first column replaced by the constant vector:

|10 1 1|

| 3 -2 1| = c1

|-1 1 -4|

Find the determinant of this c1 matrix. It will be ...

((10*-2*-4)+(1*1*-1)+(1*3*1)) - ((10*1*1)+(1*3*-4)+(1*-2*-1))

= (80 - 1 + 3) - (10 - 12 + 2) = 82 - 0 = 82

The solution for x1 is the determinant of the c1 matrix divided by the determinant of the coefficient matrix.

x1 = 82/41 = 2

To find x2, we do the same thing, only replacing the second column of the coefficient matrix by the constant vector.

| 1 10 1|

| 5 3 1| = c2

| 3 -1 -4|

The determinant of c2 is

((1*3*-4)+(10*1*3)+(1*5*-1) - (1*1*-1)+(10*5*-4)+(1*3*3))

= (-12 + 30 - 5) - (-1 - 200 + 9) = 13 + 192 = 205

The solution for x2 is the determinant of the c2 matrix divided by the determinant of the coefficient matrix.

x2 = 205/41 = 5

To find x3, we do the same thing, only replacing the second column of the coefficient matrix by the constant vector.

| 1 1 10|

| 5 -2 3| = c3

| 3 1 -1|

The determinant of c3 is

((1*-2*-1)+(1*3*3)+(10*5*1)) - ((1*3*1)+(1*5*-1)+(10*-2*3))

= (2 + 9 + 50) - (3 - 5 - 60) = 61 + 62 = 123

The solution for x3 is the determinant of the c3 matrix divided by the determinant of the coefficient matrix.

x3 = 123/41 = 3

Thus, x1=2, x2=5, x3=3.