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Problem 1
Melvin and Annie are playing a number game. Melvin tells Annie to pick a number, take half of it, and add
7 to the result.
Part A: Write an expression to represent Melvin's statement.
Part B: Use the distributive property to write an equivalent expression of the form abx + c), where a, b,
and c are constants. Then write a statement like
Melvin's to describe your new expression.

Verified Answer :

GinnyAnswer

Sure, let's walk through this step-by-step to solve the problem.

### Part A: Write an expression to represent Melvin's statement.

Melvin's statement is: "Pick a number, take half of it, and add 7 to the result."

Let's represent this mathematically:

1. Let \(\( x \)\) be the number picked by Annie.
2. Taking half of \(\( x \)\) can be represented as \(\( \frac{1}{2}x \)\) or \(\( 0.5x \)\).
3. Adding 7 to this result gives us: \(\( \frac{1}{2}x + 7 \)\) or \(\( 0.5x + 7 \)\).

Thus, the expression to represent Melvin's statement is:
\(\[ \frac{1}{2}x + 7 \]\)

### Part B: Use the distributive property to write an equivalent expression of the form abx + c, where a, b, and c are constants.

The expression provided in Part A is \(\( \frac{1}{2}x + 7 \)\) or \(\( 0.5x + 7 \)\), which is already in a linear form. In this expression:

- \(\( a = 1 \)\) (since we can rewrite the expression as \(\( 1 \cdot (0.5)x + 7 \)\))
- \(\( b = \frac{1}{2} = 0.5 \)\)
- \(\( c = 7 \)\)

Thus, the expression in the form \(\( abx + c \)\) is:
\(\[ 1 \cdot (0.5)x + 7 \]\)

### New Description

The initial description "Pick a number, take half of it, and add 7 to the result" applies perfectly to our new equivalent expression because it effectively represents the same mathematical operations.

So, the final answer for Part B with a new description is:
\(\[ (1 \cdot 0.5)x + 7 \]\)

In conclusion,
- Given the constants \(\(a=1\)\), \(\(b=0.5\)\), and \(\(c=7\)\), the new equivalent expression is \(\( (1 \cdot 0.5)x + 7 \)\).
- The new description would be: "Pick a number, multiply it by 0.5, and then add 7 to the result."

These steps ensure clarity and correctness in transforming and interpreting the original problem statement into its mathematical expression and simplified equivalent form.

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