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# Cramer'S Rule

#### notes

To use Cramer’s method, the equations are written as a1x + b1y = c1 and a2x + b2y = c2.

a_1x+b_1y=c_1   .............(I)

a_2x+b_2y=c_2  ..............(II)

Here x and y are variables, a1, b1, c1 and a2, b2, c2 are real numbers, a_1b_2-a_2b_1ne0

Now let us solve these equations.

Multiplying equation (I) by b2

a_1b_2x+b_1b_2y=c_1b_2   ............(III)

Multiplying equation (II) by b1.

a_2b_1x+b_2b_1y=c_2b_1  ..............(IV)

Subtracting equation (III) from (IV)

x=(c_1b_2-c_2b_1)/(a_1b_2-a_2b_1) ...........(V)

Similarly y=(a_1c_2-a_2c_1)/(a_b_2-a_2b_1) ...........(V)

To remember and write the expressions

c_1b_2-c_2b_1, a_1b_2-a_2b_1, a_1c_2-a_2c_1 we use the determinants.

Now a_1x+b_1y=c_1

and a_2x+b_2y=c_2

We can write 3 columns ([a_1],[a_2]), ([b_1],[b_2]), ([c_1],[c_2])

The values x, y in equation (V), (VI) are written using determinants as follows

To remember let us denote

therefore x=D_x/D, y=D_y/D

For writting D, Dx, Dy remember the order of columns ([a_1],[a_2]), ([b_1],[b_2]), ([c_1],[c_2]) From the equations,

a_1x+b_1y=c_1

and a_2x+b_2y=c_2 We get the columns ([a_1],[a_2]), ([b_1],[b_2]), ([c_1],[c_2])

• In D the column of constants ([c_1],[c_2]) is omitted.
• In Dx the column of the coefficients of x, ([a_1],[a_2])  is replaced by ([c_1],[c_2]).
• In Dy the column of the coefficients of y, ([b_1],[b_2]) is replaced by  ([c_1],[c_2]).

#### definition

Using determinants, simultaneous equaions can be solved easily and in less space. This method is known as determinant method. This method was first given by a Swiss mathematician Gabriel Cramer, so it is also known as Cramer’s method.

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