# Given that x*y=xy-3x-3y+12, verify if x*y=(x-3)(y-3)+3?

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Given x*y =xy-3x-3y+12. To verify x*y = (x-3)(y-3)+3.

Verification:

Given x*y = xy - 3x -3y +12............(1).

To verify: x*y = (x-3)(y-3)+3............(2)

Verification:

One way to verify is just expanding the right side of eq (2) and see if it tallys with the RHS of eq (1). So we from the RHS of (2): x(y-3)-3(y-3)+3 = xy-3x-3y +(-3)*(-3)+3 = xy -3x -3y +9+3 = xy-3x-3y+12 = RHS of eq(1) term by term.

Also , we can tally by putting x= 0 and y = 0. Then x*y = 0*0.3*0-3*0+12 = 12 from (1) and (0-3)(0-3)+3 = 12 from (2).

For x=3 and y=3, we get from (1) x*y = 3*3-3*3-3*3+12 = 3 and from (2) x*y = (3-3)(3-3) + 3 = 3.

For x=1 and y = 1, from (1) we get: 1*1 -3*1-3*1+12 = 7. From (2) we get: x*y = (1-3)(1-3) + 3 = (-2)(-2)+3 = 4+3 = 7.

Thus for each variable variable for more than two different values x*y returns equal. Therefore,

xy - 3x -3y +12 = (x-3)(y-3)+3 is verified and they are identities.

For

All we have to do is to demonstrate that

xy-3x-3y+12 = (x-3)(y-3)+3

For this reason, we'll open the brackets from the right side, to calculate:

xy - 3x - 3y + 12 = xy - 3x - 3y + 9 + 3

We'll add the terms 9+3, from the right side:

xy - 3x - 3y + 12 = xy - 3x - 3y + 12

The both sides have the same terms.

Another way to verify if:

xy - 3x - 3y + 12 = (x - 3)(y - 3) + 3

is to divide the expression in two parts as:

(xy - 3x - 3y + 9) + 3

And then verify if:

(xy - 3x - 3y + 9) = (x - 3)(y - 3)

to do this we factorise (xy - 3x - 3y + 9) as follows

xy - 3x - 3y + 9 = x(y - 3) - 3(y - 3)

= (x - 3)(y -3)

Thus we see that (xy - 3x - 3y + 9) = (x - 3)(y - 3)

Therefore:

xy - 3x - 3y + 12 = (x - 3)(y - 3) + 3