Question

9 decide if the following statements refer to the same or different expressions. if they are different, write an expression for each statement.
a a twice the sum of \\(x\\) and \\(y\\)
b the sum of \\(2x\\) and \\(y\\)
b a the difference between half of \\(x\\) and half of \\(y\\)
b half of the difference between \\(x\\) and \\(y\\)

Explanation:

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<reasoning>

Translate statements in part a

Using the Algebraic Modeling knowledge point:

• Statement A: "Twice the sum of \(x\) and \(y\)" translates to:

\[2(x + y) = 2x + 2y\]

• Statement B: "The sum of \(2x\) and \(y\)" translates to:

\[2x + y\]

Compare expressions in part a

• We compare the two algebraic expressions:

\[2x + 2y
eq 2x + y \quad (\text{for } y
eq 0)\]

• Since the expressions are not equivalent, they refer to different expressions.

Translate statements in part b

Using the Algebraic Modeling knowledge point:

• Statement A: "The difference between half of \(x\) and half of \(y\)" translates to:

\[\frac{1}{2}x - \frac{1}{2}y\]

• Statement B: "Half of the difference between \(x\) and \(y\)" translates to:

\[\frac{1}{2}(x - y)\]

Compare expressions in part b

• We simplify the expression from Statement B:

\[\frac{1}{2}(x - y) = \frac{1}{2}x - \frac{1}{2}y\]

• Since both statements simplify to the exact same algebraic expression, they refer to the same expression.

</reasoning>

<answer>

Question 9a

Different.

• Expression A: \(2(x + y)\)
• Expression B: \(2x + y\)

Question 9b

Same.

• Expression: \(\frac{1}{2}(x - y)\) (or \(\frac{1}{2}x - \frac{1}{2}y\))

</answer>

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Answer:

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"new_concepts": [
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<reasoning>

Translate statements in part a

Using the Algebraic Modeling knowledge point:

• Statement A: "Twice the sum of \(x\) and \(y\)" translates to:

\[2(x + y) = 2x + 2y\]

• Statement B: "The sum of \(2x\) and \(y\)" translates to:

\[2x + y\]

Compare expressions in part a

• We compare the two algebraic expressions:

\[2x + 2y
eq 2x + y \quad (\text{for } y
eq 0)\]

• Since the expressions are not equivalent, they refer to different expressions.

Translate statements in part b

Using the Algebraic Modeling knowledge point:

• Statement A: "The difference between half of \(x\) and half of \(y\)" translates to:

\[\frac{1}{2}x - \frac{1}{2}y\]

• Statement B: "Half of the difference between \(x\) and \(y\)" translates to:

\[\frac{1}{2}(x - y)\]

Compare expressions in part b

• We simplify the expression from Statement B:

\[\frac{1}{2}(x - y) = \frac{1}{2}x - \frac{1}{2}y\]

• Since both statements simplify to the exact same algebraic expression, they refer to the same expression.

</reasoning>

<answer>

Question 9a

Different.

• Expression A: \(2(x + y)\)
• Expression B: \(2x + y\)

Question 9b

Same.

• Expression: \(\frac{1}{2}(x - y)\) (or \(\frac{1}{2}x - \frac{1}{2}y\))

</answer>

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