Questions: For each expression, select all equivalent expressions from the list. (a) x+x+x: O x+3, O 3x, O 2x-x, O 3+x (b) 24y-16: O 8(3y-2), O 8y, O 12y-8, O 8 · 3y-8 · 2

For each expression, select all equivalent expressions from the list. (a) x+x+x: O x+3, O 3x, O 2x-x, O 3+x (b) 24y-16: O 8(3y-2), O 8y, O 12y-8, O 8 · 3y-8 · 2
Transcript text: For each expression, select all equivalent expressions from the list. \begin{tabular}{|l|llll|} \hline (a) $x+x+x$ & $O x+3$ & $O 3 x$ & $O 2 x-x$ & $O 3+x$ \\ \hline (b) $24 y-16$ & $O 8(3 y-2)$ & $O 8 y$ & $O 12 y-8$ & $O 8 \cdot 3 y-8 \cdot 2$ \\ \hline \end{tabular}
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Solution

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Solution Steps

To determine which expressions are equivalent, we need to simplify each given expression and compare it to the options provided.

For (a) \(x + x + x\):

  1. Simplify the expression to \(3x\).
  2. Compare \(3x\) with each option to find the equivalent ones.

For (b) \(24y - 16\):

  1. Factor out the greatest common factor (GCF) from the expression.
  2. Simplify the factored expression.
  3. Compare the simplified expression with each option to find the equivalent ones.
Step 1: Simplify the Expression \(x + x + x\)

The given expression is \(x + x + x\). Simplifying this, we get: \[ x + x + x = 3x \]

Step 2: Compare with Options for (a)

We compare \(3x\) with the given options:

  • \(x + 3\)
  • \(3x\)
  • \(2x - x\)
  • \(3 + x\)

Simplifying each option:

  • \(x + 3\) remains \(x + 3\)
  • \(3x\) remains \(3x\)
  • \(2x - x\) simplifies to \(x\)
  • \(3 + x\) remains \(3 + x\)

Only \(3x\) is equivalent to \(3x\).

Step 3: Simplify the Expression \(24y - 16\)

The given expression is \(24y - 16\). Factoring out the greatest common factor (GCF) of 8, we get: \[ 24y - 16 = 8(3y - 2) \]

Step 4: Compare with Options for (b)

We compare \(24y - 16\) with the given options:

  • \(8(3y - 2)\)
  • \(8y\)
  • \(12y - 8\)
  • \(8 \cdot 3y - 8 \cdot 2\)

Simplifying each option:

  • \(8(3y - 2)\) simplifies to \(24y - 16\)
  • \(8y\) remains \(8y\)
  • \(12y - 8\) remains \(12y - 8\)
  • \(8 \cdot 3y - 8 \cdot 2\) simplifies to \(24y - 16\)

Both \(8(3y - 2)\) and \(8 \cdot 3y - 8 \cdot 2\) are equivalent to \(24y - 16\).

Final Answer

For (a), the equivalent expression is: \[ \boxed{3x} \]

For (b), the equivalent expressions are: \[ \boxed{8(3y - 2) \text{ and } 8 \cdot 3y - 8 \cdot 2} \]

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