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 [tex]\( x \)[/tex] be the number picked by Annie.
2. Taking half of [tex]\( x \)[/tex] can be represented as [tex]\( \frac{1}{2}x \)[/tex] or [tex]\( 0.5x \)[/tex].
3. Adding 7 to this result gives us: [tex]\( \frac{1}{2}x + 7 \)[/tex] or [tex]\( 0.5x + 7 \)[/tex].
Thus, the expression to represent Melvin's statement is:
[tex]\[ \frac{1}{2}x + 7 \][/tex]
### 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 [tex]\( \frac{1}{2}x + 7 \)[/tex] or [tex]\( 0.5x + 7 \)[/tex], which is already in a linear form. In this expression:
- [tex]\( a = 1 \)[/tex] (since we can rewrite the expression as [tex]\( 1 \cdot (0.5)x + 7 \)[/tex])
- [tex]\( b = \frac{1}{2} = 0.5 \)[/tex]
- [tex]\( c = 7 \)[/tex]
Thus, the expression in the form [tex]\( abx + c \)[/tex] is:
[tex]\[ 1 \cdot (0.5)x + 7 \][/tex]
### 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:
[tex]\[ (1 \cdot 0.5)x + 7 \][/tex]
In conclusion,
- Given the constants [tex]\(a=1\)[/tex], [tex]\(b=0.5\)[/tex], and [tex]\(c=7\)[/tex], the new equivalent expression is [tex]\( (1 \cdot 0.5)x + 7 \)[/tex].
- 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.
