Answer:
[tex]\frac{a}{(a-c)\cdot (b-c)}[/tex]
Step-by-step explanation:
We must use algebraic means to simplify the equation given. The procedure is presented below:
1) [tex]\frac{a}{(a-b)\cdot (a-c)} + \frac{b}{(b-c)\dot (b-a) } + \frac{c}{(a-c)\cdot (b-c)}[/tex] Given.
2) [tex]\frac{a}{(a-b)\cdot (a-c)} + \frac{b}{-(a-b)\cdot (b-c)} + \frac{c}{(a-c)\cdot (b-c)}[/tex] Commutative property/Distributive property/[tex](-1)\cdot a = -a[/tex]/[tex](-1)\cdot (-a) = a[/tex]
3) [tex]\frac{a}{(a-b)\cdot (a-c)} + \frac{(-b)}{(a-b)\cdot (b-c)} + \frac{c}{(a-c)\cdot (b-c)}[/tex] [tex]-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}[/tex]
4) [tex]\frac{a\cdot (b-c)}{(a- b)\cdot (a-c)\cdot (b-c)} + \frac{(-b)\cdot (a-c)}{(a-b)\cdot (b-c)\cdot (a-c)} + \frac{c\cdot (a-b)}{(a-c)\cdot (b-c)\cdot (a-b)}[/tex] Modulative property/Existence of aditive inverse/Definition of division
5) [tex]\frac{a\cdot (a-c) + (-b)\cdot (a-c)+c\cdot (a-b)}{(a-b)\cdot (a-c)\cdot (b-c)}[/tex] Distributive property/Definition of division
6) [tex]\frac{a^{2}-a\cdot c -a\cdot b + b\cdot c+a\cdot c-b\cdot c}{(a-b)\cdot (a-c)\cdot (b-c)}[/tex] Distributive and commutative properties/[tex](-a) \cdot b = -a\cdot b[/tex]/[tex](-a)\cdot (-b) = a\cdot b[/tex]/Definition of power
7) [tex]\frac{a^{2}-a\cdot b}{(a-b)\cdot (a-c)\cdot (b-c)}[/tex] Commutative, associative and modulative properties/Existence of additive inverse
8) [tex]\frac{a\cdot (a-b)}{(a-b)\cdot (a-c)\cdot (b-c)}[/tex] Commutative property
9) [tex]\frac{a}{(a-c)\cdot (b-c)}[/tex] Commutative and associative properties/Existence of multiplicative inverse/Result
