For all non-zero real numbers a, b, and c, which of the following is equivalent to the exp… — ACT

For all non-zero real numbers $a$ , $b$ , and $c$ , which of the following is equivalent to the expression $\frac{a(b - c) - b(a - c)}{c}$ ?

b - a

Applying the distributive property to the numerator yields

ab - ac - ba + bc

. Since multiplication is commutative,

ba = ab

, allowing us to cancel the

ab

and

-ba

terms. This leaves

bc - ac

in the numerator. Factoring out

c

gives

c(b - a)

. Dividing the entire numerator by the denominator

c

simplifies the expression to

b - a

Apply the distributive property to expand the terms in the numerator.

a(b - c) - b(a - c) = ab - ac - ba + bc

Distribute

a

to both

b

and

-c

, and distribute

-b

to both

a

and

-c

. Remember that multiplying two negative numbers yields a positive result, so

-b \times -c = bc

Combine like terms in the expanded numerator.

ab - ac - ab + bc = bc - ac

Since multiplication is commutative,

ba = ab

. Thus, the terms

ab

and

-ab

cancel each other out to zero.

Factor out the common term

c

from the numerator and divide by the denominator

c

\frac{c(b - a)}{c} = b - a

Factoring out

c

from

bc - ac

gives

c(b - a)

. Dividing by

c

cancels the common factor in the numerator and denominator, leaving the simplified expression.

Order of Operations and Number Properties

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