Question

for each expression, select all equivalent expressions from the list.
(a) $11x - 3x + 7x$ $square$ $8x + 7x$ $square$ $8x - 7x$ $square$ $15 + x$ $square$ $15x$
(b) $4(1 + 6y)$ $square$ $4 + 1\cdot4 + 6y$ $square$ $4 + 24y$ $square$ $4\cdot1 + 4\cdot6y$ $square$ $4 + 6y$

Explanation:

Part (a) Step 1: Simplify \(11x - 3x + 7x\)

First, combine like terms: \(11x - 3x = 8x\), then \(8x + 7x = 15x\). Also, \(11x - 3x\) is \(8x\), so \(8x + 7x\) is equivalent to the original expression.

Part (a) Step 2: Check each option

• \(8x + 7x\): As shown, \(11x - 3x = 8x\), so \(8x + 7x\) is equivalent.
• \(8x - 7x\): This is \(x\), not equivalent.
• \(15 + x\): This has a constant and \(x\), not equivalent to \(15x\).
• \(15x\): Simplifying the original gives \(15x\), so this is equivalent.

Part (b) Step 1: Apply the distributive property to \(4(1 + 6y)\)

The distributive property states \(a(b + c)=ab + ac\). So \(4(1 + 6y)=4\times1 + 4\times6y = 4 + 24y\).

Part (b) Step 2: Check each option

• \(4 + 1\cdot4 + 6y\): This simplifies to \(4 + 4 + 6y = 8 + 6y\), not equivalent.
• \(4 + 24y\): This is the result of distributing, so equivalent.
• \(4\cdot1 + 4\cdot6y\): This is the application of the distributive property, so equivalent.
• \(4 + 6y\): This is missing the multiplication of 4 and 6y, not equivalent.

Answer:

(a) Equivalent expressions: \(8x + 7x\), \(15x\)
(b) Equivalent expressions: \(4 + 24y\), \(4\cdot1 + 4\cdot6y\)