Question

1. which of the following expressions are equivalent to the expression -5(y - 2y) + y, given for all values of y

a. 2(2y + 2y) - 2y
b. 7y - 4y×2y
c. 3(2y)
d. 6y
e. -y + 5(y - 2y)

Explanation:

Step1: Simplify the original expression

First, simplify the expression inside the parentheses: \( y - 2y=-y \). Then the original expression \(-5(y - 2y)+y\) becomes \(-5(-y)+y\). Multiply \(-5\) and \(-y\) to get \(5y\), so the expression is \(5y + y\). Combine like terms: \(5y + y = 6y\).

Step2: Analyze Option A

Simplify \(2(2y + 2y)-2y\). First, add inside the parentheses: \(2y+2y = 4y\). Then multiply by 2: \(2\times4y = 8y\). Subtract \(2y\): \(8y-2y = 6y\). This matches the simplified original expression.

Step3: Analyze Option B

Simplify \(7y-4y\times2y\). First, multiply \(4y\) and \(2y\) to get \(8y^{2}\). So the expression is \(7y - 8y^{2}\), which is not equivalent to \(6y\) (it has a quadratic term).

Step4: Analyze Option C

Simplify \(3(2y)=6y\). This matches the simplified original expression.

Step5: Analyze Option D

The expression is \(6y\), which matches the simplified original expression.

Step6: Analyze Option E

Simplify \(-y + 5(y - 2y)\). First, simplify inside the parentheses: \(y-2y=-y\). Then multiply by 5: \(5\times(-y)=-5y\). Now the expression is \(-y-5y=-6y\), which is not equivalent to \(6y\).

Answer:

A. \(2(2y + 2y)-2y\), C. \(3(2y)\), D. \(6y\)