Question

for each expression, select all equivalent expressions from the list
(a) ( 7x - 2x + 8x ) ( square 5x + 8x ) ( square 8x - 5x ) ( square 13 + x ) ( square 13x )
(b) ( 7(1 + 6y) ) ( square 7 + 6y ) ( square 7 cdot 1 + 7 cdot 6y ) ( square 7 + 42y ) ( square 7 + 1 cdot 7 + 6y )

Explanation:

Step1: Solve part (a)

First, simplify \(7x - 2x + 8x\). Combine like terms: \( (7 - 2 + 8)x = 13x\). Now check each option:

• \(5x + 8x = 13x\), so this is equivalent.
• \(8x - 5x = 3x\), not equivalent.
• \(13 + x\) is not equivalent (has a constant term and different variable combination).
• \(13x\) is equivalent (matches our simplified form).

Step2: Solve part (b)

Simplify \(7(1 + 6y)\) using the distributive property \(a(b + c)=ab + ac\). So \(7\times1 + 7\times6y = 7 + 42y\). Check each option:

• \(7 + 6y\) is not equivalent.
• \(7\cdot1 + 7\cdot6y\) is the distributive form, which simplifies to \(7 + 42y\), so this is equivalent.
• \(7 + 42y\) is the simplified form, so this is equivalent.
• \(7 + 1\cdot7 + 6y = 7 + 7 + 6y = 14 + 6y\), not equivalent.

Answer:

(a) Equivalent expressions: \(5x + 8x\), \(13x\)
(b) Equivalent expressions: \(7\cdot1 + 7\cdot6y\), \(7 + 42y\)