solve 3x3 matrix w/ cramers rule?

Question

please help! I have zero idea of how to do this

4x-2y+3z=14
2x+y-5x=5
-3x-2y+5z=-7

Answer 1

Cramer's Rule uses two determinants, in a fraction. The bottom is the determaiant formed by the coefficients of the variables, and the top is the determinant formed by taling the same coefficients but replacing the column corresponding to the variable you're trying to find with the answers. Here for example to find x, put det

14 -2 3
5 1 -5 (I assume you meant -5z not -5x)
7 -2 5

over det

4 -2 3
2 1 -5
-3 -2 5

Answer 2

In matrix algebra this would be written as

[4 -2 3] [x] [14]
|2 1 -5] [y] = [5]
[-3 -2 5] [z] [-7]

Cramer's rule tells you the solution for each are the determinant with the constant vector substituted into the variable's column, over the determinant of the matrix itself. That is,


x =
|14 -2 3|
| 5 1 -5|
|-7 -2 5|
_________
|4 -2 3|
|2 1 -5|
|-3 -2 5|

= (14(5 - 10) + 2(25 - 35) + 3(-10 + 7))/(4(5 - 10) + 2(10 - 15) + 3(-4 + 4))
= (-70 - 20 - 9)/(-20 -10 + 0) = -99/-30 = 33/10.

to solve for y, get the determinant with the vector (14,5,-7) as the second column, over 30; for z, the determinant where the vector is the 3rd column, over 30. (30 is the determinant of the original matrix)

If you don't know how to find the determinant of a 3 x 3 matrix, use

|a b c|
|d e f | = a(ei-hf) -b(di - gf) + c(dh - ge)
|g h i |

Answer 3

that's not the matrix despite the fact that it quite is something! take the 1rst and the third eq 4x + 5y + z = 6 (mult by skill of (-2)) x + 2y + 2z = 6 -8x - 10y - 2z = -12 x + 2y + 2z = 6 upload the two eq -8x + x - 10y +2y - 2z + 2z = -6 -7x - 8y = -6 (a million) take the 2nd and the third eq 2x - y + 2z =11 x + 2y + 2z = 6 (MULT by skill of (-a million) 2x - y + 2z =11 -x - 2y - 2z = -6 upload thw 2 eq 2x - x - y - 2y + 2z - 2z = 11 - 6 x - 3y = 5 (2) Now take (a million) & (2) -7x - 8y = -6 x - 3y = 5 (Mult by skill of 7) -7x - 8y = -6 7x - 21y = 35 upload the eq. -8y - 21y = -6 + 35 - 29y = 29 y = -a million So, use y = -a million to (a million) or (2). enable's % x - 3y = 5 x - 3(-a million) = 5 x +3 = 5 x = 2 Now, use y = -a million & x = 2 to between the preliminary eq to discover z 4x + 5y +z = 6 4*2 +5*(-a million) + z = 6 8 - 5 + z = 6 z = 3