Question

[Phương Nguyễn]

a)Chứng minh đẳng thức :

a) (x-y)-(x-z)=(z+x)- (y+x)

b) (x-y+z)-(y+z-x)-(x-y)= (z-y)-(z-x)

c) a(b+c)-b(a-c)=(a+b) c

Answer 1

[Hoàng Thị Ngọc Anh]

a) \(\left(x-y\right)-\left(x-z\right)=\left(z+x\right)-\left(y+x\right)\)

BL:

Ta có: \(\left(x-y\right)-\left(x-z\right)\)

\(=x-y-x+z\)

\(=z+x-y-x\)

\(=\left(z+x\right)-\left(y+x\right)\)

\(\Rightarrow\) \(\left(x-y\right)-\left(x-z\right)=\left(z+x\right)-\left(y+x\right)\)

b) \(\left(x-y+z\right)-\left(y+z-x\right)-\left(x-y\right)=\left(z-y\right)-\left(z-x\right)\)

BL:

Lại có: \(\left(x-y+z\right)-\left(y+z-x\right)-\left(x-y\right)\)

\(=x-y+z-y-z+x-x+y\)

\(=\left(x-y-x+y\right)+\left(z-y\right)-\left(z-x\right)\)

\(=\left(z-y\right)-\left(z-x\right)\)

\(\Rightarrow\) \(\left(x-y+z\right)-\left(y+z-x\right)-\left(x-y\right)=\left(z-y\right)-\left(z-x\right)\)

c) \(a\left(b+c\right)-b\left(a-c\right)=\left(a+b\right)c\) BL: Ta lại có: \(a\left(b+c\right)-b\left(a-c\right)=\left(a+b\right)c\) \(=ab+ac-ba+bc\) \(=\left(ab-ba\right)+\left(ac+bc\right)\) \(=0+\left(a+b\right)c\) \(=\left(a+b\right)c\) \(\Rightarrow\) \(a\left(b+c\right)-b\left(a-c\right)=\left(a+b\right)c\) \(\rightarrow\) đpcm.