Question

for each expression, select all equivalent expressions from the list.
(a) $6(8x - 5)$ $square$ $48x - 5$ $square$ $6\cdot8x - 6\cdot5$ $square$ $14x - 5$ $square$ $48x - 30$
(b) $10y + 6y + 5y$ $square$ $10y + 11y$ $square$ $11y - 10y$ $square$ $21y$ $square$ $21 + y$

Explanation:

Part (a)

Step1: Apply Distributive Property

The distributive property states that \( a(b - c)=a\cdot b - a\cdot c \). For \( 6(8x - 5) \), we distribute 6 to both terms inside the parentheses:
\( 6(8x - 5)=6\cdot8x-6\cdot5 \)

Step2: Simplify the Products

Calculate \( 6\cdot8x = 48x \) and \( 6\cdot5 = 30 \). So, \( 6(8x - 5)=48x - 30 \). Also, the expression \( 6\cdot8x - 6\cdot5 \) is just the intermediate step using the distributive property, which is equivalent to the original expression before simplifying the products.

Part (b)

Step1: Combine Like Terms

Like terms are terms with the same variable raised to the same power. For \( 10y+6y + 5y \), we add the coefficients: \( 10 + 6+5 = 21 \), so \( 10y+6y + 5y=21y \). Also, \( 10y + 11y=10y+(6y + 5y)=10y + 6y+5y \) (since \( 6y + 5y = 11y \)), so \( 10y+6y + 5y=10y + 11y \).

Answer:

(a) Equivalent expressions: \( 6\cdot8x - 6\cdot5 \), \( 48x - 30 \)
(b) Equivalent expressions: \( 10y + 11y \), \( 21y \)