Question

Ax+by=(a-b) , bx - ay =(a+b) simultaneous linear equations using cross multiplication

Answer 1

`x=1\,,\,y=-1`

Explanation:

Given:
`ax+by=a-b\,,\,bx-ay=a+b`

To solve: the given linear equations

Solution:

Consider the equations:

`A_1x+B_1y+C_1=0\\A_2x+B_2y+C_2=0`

By method of cross multiplication:

`(x)/(B_1C_2-B_2C_1)=(y)/(C_1A_2-C_2A_1)=(1)/(A_1B_2-A_2B_1)`

For equations:
`ax+by=a-b\,,\,bx-ay=a+b`

`ax+by-(a-b)=0\\bx-ay-(a+b)=0`

Take
`A_1=a\,,\,B_1=b\,,\,C_1=-(a-b)\,,\,A_2=b\,,\,B_2=-a\,,\,C_2=-(a+b)`

So,

`(x)/(-b(a+b)-a(a-b))=(y)/(-b(a-b)+a(a+b))=(1)/(-a^2-b^2)\\(x)/(-ab-b^2-a^2+ab)=(y)/(-ab+b^2+a^2+ab)=(1)/(-(a^2+b^2))\\(x)/(-(a^2+b^2))=(y)/(a^2+b^2)=(1)/(-(a^2+b^2))\\(x)/(-(a^2+b^2))=(1)/(-(a^2+b^2))\,,\,(y)/(a^2+b^2)=(1)/(-(a^2+b^2))\\x=1\,,\,y=-1`