Use Cramer's Rule to solve (if possible) the system of equations.

Question

-2x +3y -5z = -11

4x -y +z = -3

-x -4y +6z =15

It would help a lot if you could show your work. Thanks!

Answer 1

Create matrix -2 3 -5 and find the determinant D = -2(-6+4) -3*(24+1) -5*(-16-1) = 14

4 -1 1

-1 -4 6

Create new matrix for x by substituing the solution set into column 1 -11 3 -5 , then find determinant.

-3 -1 1

15 -4 6

det =-11*(-6+4)-3*(-18-15) -5*(12+15) = -14

Calculate the x value by dividing -14/D = -1

Repeat for y & z finding y=4 and z=5

Answer 2

The coefficient matrix A = -2 3 -5 b = -11

4 -1 1 -3

-1 -4 6 15

det(A) = (-2)(-1)(6) + (4)(-4)(-5) + (3)(1)(-1) - (-5)(-1)(-1) -(1)(-4)(-2) - (6)(4)(3) = 12 + 80 -3 + 5 -8 -72 = 14

Now replace the first column of A with b.

Ax = -11 3 -5 Find det(Ax) = (-11)(-1)(6) + (3)(1)(15) + (-5)(-3)(-4) - (-5)(-1)(15) - (-3)(3)(6)

-3 -1 1 - (-11)(-4)(1) = 66 + 45 -60 -75 + 54 -44 = -14

15 -4 6

Then x = det(Ax)/det(A) = -14/14 = -1

Similarly

Ay = -2 -11 -5

4 -3 1

-1 15 6

(and then y = det(Ay)/det(A))

See if you can find y and z yourself.