Question

for each expression, select all equivalent expressions from the list.
(a) $9x + 81$
$\square$ $9(x + 9)$
$\square$ $9 \cdot x + 9 \cdot 9$
$\square$ $90x$
$\square$ $9(9x + 1)$
(b) $15 + 10y - 1 - y$
$\square$ $14y + 9$
$\square$ $14y + 9y$
$\square$ $9 + 14y$
$\square$ $9y + 14$

Explanation:

Part (a) - Simplify \(9x + 81\)

Step 1: Factor out the greatest common factor (GCF)

The GCF of \(9x\) and \(81\) is \(9\). We can factor \(9\) out of each term:

$$ 9x + 81 = 9(x) + 9(9) = 9(x + 9) $$

Step 2: Check other expressions

• \(9 \cdot x + 9 \cdot 9\): This is the expanded form of \(9(x + 9)\) (by the distributive property), so it is equivalent.
• \(9(9x + 1)\): Let's expand this: \(9(9x + 1) = 81x + 9\), which is not equivalent to \(9x + 81\).
• \(90x\): This is a different expression with a coefficient of \(90\) for \(x\), not equivalent.

Part (b) - Simplify \(15 + 10y - 1 - y\)

Step 1: Combine like terms

First, combine the constant terms (\(15 - 1\)) and the \(y\)-terms (\(10y - y\)):

$$ 15 + 10y - 1 - y = (15 - 1) + (10y - y) = 14 + 9y $$

We can also write this as \(9y + 14\) (by the commutative property of addition).

Step 2: Check other expressions

• \(14y + 9\): This has \(14y\) and \(9\), but we have \(9y\) and \(14\), so not equivalent.
• \(9 + 14y\): Similar to the above, not equivalent.
• \(14y + 9y\): This combines to \(23y\), not equivalent.
• \(9y + 14\): This is the simplified form we found, so it is equivalent.

Answer:

(a) The equivalent expressions to \(9x + 81\) are \(9(x + 9)\) and \(9 \cdot x + 9 \cdot 9\).
(b) The equivalent expression to \(15 + 10y - 1 - y\) is \(9y + 14\).

(To mark the checkboxes: For part (a), check \(9(x + 9)\) and \(9 \cdot x + 9 \cdot 9\). For part (b), check \(9y + 14\).)