Lesson Cramer's Rule

This Lesson (Cramer's Rule) was created by by longjonsilver(2297)

: View Source, Show
About longjonsilver: I have a new job in September, teaching

How do we solve the 3 equations:

x+2y+3z = 14
4x-5y+6z = 32
7x+8y-9z = 50

Well, we can pick one of the variables in one of the equations and then substitute that into the other 2 equations and then solve those two equations simultaneously. A rather long winded algebraic solution.

A much better way, is to use the theory of determinants in matrices to help. This is Cramer's Rule.

First, rewrite the system of equations in a matrix form:

Find the determinant, D, of the matrix %28+matrix%28+3%2C3%2C1%2C2%2C3%2C4%2C-5%2C6%2C7%2C8%2C-9+%29%29+

D=det+%28matrix%28+3%2C3%2C1%2C2%2C3%2C4%2C-5%2C6%2C7%2C8%2C-9+%29%29+

.

Then put the column %28matrix%283%2C1%2C14%2C32%2C50%29%29

in place of the 1st, 2nd and 3rd columns sequentially, and calculate determinants as follows:

Dx+=+det+%28matrix%283%2C3%2C14%2C2%2C3%2C32%2C-5%2C6%2C50%2C8%2C-9%29%29

Dx+=+det+%28matrix%283%2C3%2C14%2C2%2C3%2C32%2C-5%2C6%2C50%2C8%2C-9%29%29

Dy+=+det+%28matrix%283%2C3%2C1%2C14%2C3%2C4%2C32%2C6%2C7%2C50%2C-9%29%29

Dz+=+det+%28matrix%283%2C3%2C1%2C2%2C14%2C4%2C-5%2C32%2C7%2C8%2C50%29%29

.

Cramer's Rule says:

x = Dx/D

y = Dy/D

z = Dz/D

Working these out, gives:

D   =   354
Dx = 2652
Dy =   432
Dz =   480

x = 2652/354 = 7.4915257
y =   432/354 = 1.2203389
z =   480/354 = 1.3559322

An important note: surely, this method assumes that determinant D is not equal to zero.

This lesson has been accessed 15625 times.