Answer:
I)
[tex]\displaystyle A + B + C = 2x^2 + 4xy + 2[/tex]
II)
[tex]\displaystyle A - B + C = 6x^2 -8xy[/tex]
III)
[tex]\displaystyle A + B - C= -4x^2 + 10xy -12[/tex]
Step-by-step explanation:
We are given the three equations:
[tex]\displaystyle A = x^2 + xy -6,\, B = 6xy - 2x^2 +1 \text{ and } C = 3x^2 + 7 -3xy[/tex]
I)
We want to find:
[tex]\displaystyle A + B + C[/tex]
Substitute:
[tex]\displaystyle = (x^2 + xy - 6) + (6xy -2x^2 + 1) + (3x^2 + 7 -3xy)[/tex]
Rewrite:
[tex]\displaystyle = (x^2 + 3x^2 - 2x^2) + (xy + 6xy - 3xy) + (-6 + 1 + 7)[/tex]
And combine like terms. Hence:
[tex]\displaystyle A + B + C = 2x^2 + 4xy + 2[/tex]
II)
We want to find:
[tex]\displaystyle A - B + C[/tex]
Likewise, substitute:
[tex]\displaystyle = (x^2 + xy - 6) - (6xy - 2x^2 + 1) + (3x^2 + 7 - 3xy)[/tex]
Distribute:
[tex]\displaystyle = (x^2 + xy - 6) + (-6xy +2x^2 - 1) + (3x^2 + 7 - 3xy)[/tex]
Rewrite:
[tex]\displaystyle = (x^2 + 2x^2 +3x^2) + (xy - 6xy -3xy) + (-6 -1 + 7 )[/tex]
And combine like terms. Hence:
[tex]\displaystyle A - B + C = 6x^2 -8xy[/tex]
III)
We want to find:
[tex]\displaystyle A + B - C[/tex]
Substitute:
[tex]\displaystyle = (x^2 + xy - 6) + (6xy - 2x^2 + 1) - (3x^2 + 7 - 3xy)[/tex]
Distribute:
[tex]\displaystyle = (x^2 + xy - 6) + (6xy - 2x^2 + 1) + (-3x^2 - 7 + 3xy)[/tex]
Rewrite:
[tex]\displaystyle = (x^2 - 2x^2 - 3x^2) + (xy +6xy +3xy) + (-6 +1 - 7)[/tex]
And combine like terms. Hence:
[tex]\displaystyle A + B - C= -4x^2 + 10xy -12[/tex]
