For all non-zero real numbers x, y, and z, which of the following mathematical equations m… — ACT

For all non-zero real numbers $x$ , $y$ , and $z$ , which of the following mathematical equations must be true?

A
$x - (y + z) = x - y + z$
B
$x(y - z) = xy - z$
C
$\frac{z}{x + y} = \frac{z}{x} + \frac{z}{y}$
$\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}$ Answer
E
$x \div (y \div z) = (x \div y) \div z$

Answer

The equation

\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

must be true.

The equation stating that a sum in the numerator divided by a single term in the denominator can be split into two separate fractions is correct because division distributes over addition from the right side. This can be verified by rewriting the division as multiplication by the reciprocal:

\frac{x + y}{z} = (x + y) \cdot \frac{1}{z} = x \cdot \frac{1}{z} + y \cdot \frac{1}{z} = \frac{x}{z} + \frac{y}{z}

Step-by-Step Solution

Analyze the properties of division and addition in the equation

\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

The division of a sum by a number can be rewritten as the multiplication of the sum by the reciprocal of that number:

(x + y) \cdot \frac{1}{z}

To apply the distributive property of multiplication over addition.

Distribute the term

\frac{1}{z}

to both

x

and

y

(x + y) \cdot \frac{1}{z} = x \cdot \frac{1}{z} + y \cdot \frac{1}{z} = \frac{x}{z} + \frac{y}{z}

The distributive property states that

a(b + c) = ab + ac

for any real numbers

a

b

, and

c

Confirm that this identity holds for all non-zero real numbers

x

y

, and

z

Since

z \neq 0

, the denominators are defined, and the relation

\frac{x + y}{z} = \frac{x}{z} + \frac{y}{z}

is always true.

To establish that the statement must be true under the given conditions.

Key Concept

Distributive Property of Real Numbers

Alternative Method

We can plug in simple numbers (e.g.,

x = 2

y = 3

z = 4

) to test each option and eliminate those that do not produce a true statement.

Estimated Time:1m 0s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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