#### notes

To use Cramer’s method, the equations are written as a_{1}x + b_{1}y = c1 and a_{2}x + b_{2}y = c_{2}.

`a_1x+b_1y=c_1` .............(I)

`a_2x+b_2y=c_2` ..............(II)

Here x and y are variables, a_{1}, 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 b_{2}

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

Multiplying equation (II) by b_{1}.

`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 D
_{x}the column of the coefficients of x, `([a_1],[a_2])` is replaced by `([c_1],[c_2])`. - In D
_{y}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**.