What is Cramer's Rule?

Question

What is Cramer's rule and how do i use it?
EX:
x+2y=5
2x-y=-3

Answer 1

Cramer's Rule is an algebraic method that allows you by means of the matrices to solve a linear equation system:

So first you build your matrices, X and Y:

X =
| 5 2 | ==> Dtrmt(X) = (5)(-1) - (2)(-3) = -5 + 6 = 1
| -3 -1 |

Y =
| 1 5 | ==> Dtrmt(Y) = (1)(-3) - (2)(5) = -3 - 10 = -13
| 2 -3 |

X is made of the replacement of its column (1st one) by the
column of independent terms. (five and minus three).
I mean, the first column was replaced.

Y is made of the replacement of its column (2nd one) by the
column of independent terms. (five and minus three).
I mean, the second column was replaced.

We found C =
| 1 2 | ==> Dtrmt(C) = (1)(-1) - (2)(2) = -1 - 4 = -5
| 2 -1 |

For C, you replace nothing.

So, the solution is:

x = Dtrmt(X) / Dtrmt(C) = 1/(-5) = -1/5 = -.2
y = Dtrmt(Y) / Dtrmt(C) = (-13)/(-5) = 13/5 = 2 3/5 = 2.6

If the degree of the system would be higher you should replace the column of the variable by the the columns of the independent terms.
You divide this determinant (matrix delta) by the independent terms'.

Hope it would be useful. I love Math. It is nice to remember it.

Answer 2

The last poster almost got it right, but both x and y should be the negative of his answers.

For two equations in two unknowns, the proceedure goes as follows:
The determinant of a matrix
[a b]
[c d]
is defined to be ad-bc.

if you start with a system
ax+by=g
cx+dy=h,
then the value of x may be obtained by
1) replacing the column associated with x by the constant column and taking the determinant:
[g b]
[h d]
giving gd-hb
2) dividing this by the determinant of the coefficient matrix and then taking the negative:
[a b]
[c d]
with determinant ad-bc,
so x=-(gd-hb)/(ad-bc).

The value of y may be obtained by the same proceedure, except that the y column is replaced by the constants column in step 1.

This proceedure can be generalized to any system of n equations in n unknowns by appropriately defining the determinant of an nxn matrix. Cramer's rule is the statement that this always works as long as the determinant of the coefficient matrix is not zero. If it is, the system is inderterminate. This means there are either no solutions or infinitely many solutions.

The problem with Cramer's rule for larger systems is that computing the determinants is time consuming. There are faster methods of solving such systems, so Cramer's rule tends to have mainly theoretical value.

Answer 3

cramer's rule is used to solve simultaneous linear equations. to use the rule you should know to solve DETERMINANTS.if u r an indian buy an 11th std science stream textbook

Answer 4

A theorem in linear algebra for solving simultaneous linear equations using determinents.

Answer 5

To Bust into Jerry's apartment without ever knocking

Answer 6

2 points.