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

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

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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.

 

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giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

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.

krishna-agrawala's profile pic

krishna-agrawala | College Teacher | (Level 3) Valedictorian

Posted on

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

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