For all non-zero real numbers x and y, which of the following is equivalent to the express… — ACT

For all non-zero real numbers $x$ and $y$ , which of the following is equivalent to the expression below?

x - y[x - (y - x)] - y^2

A
$2x^2 - 3xy$
B
$x - 2xy - 2y^2$
$x - 2xy$ Answer
D
$x$
E
$x - 2y^2$

Answer

The correct answer is

x - 2xy

The correct answer is

x - 2xy

. We begin by simplifying the innermost grouping symbols first. The expression inside the parentheses is

x - (y - x) = x - y + x = 2x - y

. Substituting this back into the expression gives

x - y[2x - y] - y^2

. Next, we distribute the factor of

-y

to the terms inside the brackets, yielding

-y(2x) - y(-y) = -2xy + y^2

. The expression then becomes

x - 2xy + y^2 - y^2

. Finally, combining the like terms

+y^2

and

-y^2

cancels them out, resulting in the simplified expression

x - 2xy

Step-by-Step Solution

Simplify the expression inside the innermost parentheses:

x - (y - x)

2x - y

Distribute the negative sign to the terms inside the parentheses:

-(y - x) = -y + x

. Combining like terms gives

x + x - y = 2x - y

Substitute the simplified expression back into the main expression:

x - y[2x - y] - y^2

x - y[2x - y] - y^2

This updates the expression to show the next level of grouping symbols to resolve.

Distribute the

-y

to the terms inside the brackets:

-y[2x - y]

-2xy + y^2

Apply the distributive property:

-y \times 2x = -2xy

, and

-y \times (-y) = y^2

Combine the distributed terms back into the expression and simplify:

x - 2xy + y^2 - y^2

x - 2xy

The

+y^2

and

-y^2

terms cancel each other out, leaving the simplified expression

x - 2xy

Key Concept

Applying the order of operations (PEMDAS) to evaluate expressions with nested grouping symbols, and correctly distributing positive and negative signs.

Estimated Time:1m 30s

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