Nancy is thinking of a number. If you multiply it by 6 and add 9 , you get 75 . (a) Could Nancy's number be 9 ? Why or why not? (b) What techniques could you use to find Nancy's number? (c) What is Nancy's number? (a) Could Nancy's number be 9 ? Why or why not? Nancy's number (b) What techniques could you use to find Nancy's number? Select all that apply. \( \square \) A. Guess and check. \( \square \) B. Eliminate possibilities. \( \square \) c. Make an orderly list or table. \( \square \) D. Use algebra. (c) What is Nancy's number? \( \square \)

Let's break down the problem step by step. ### Known Conditions 1. Nancy's number is unknown, let's denote it as \( x \). 2. The equation given is: \( 6x + 9 = 75 \). ### (a) Could Nancy's number be 9? Why or why not? To determine if Nancy's number could be 9, we can substitute \( x = 9 \) into the equation and see if it holds true. Calculating: \[ 6(9) + 9 = 54 + 9 = 63 \] Since \( 63 \neq 75 \), Nancy's number cannot be 9. ### (b) What techniques could you use to find Nancy's number? The techniques that could be used to find Nancy's number include: - \( \square \) A. Guess and check. - \( \square \) B. Eliminate possibilities. - \( \square \) C. Make an orderly list or table. - \( \square \) D. Use algebra. All of these techniques can be useful, but using algebra is the most straightforward method. ### (c) What is Nancy's number? To find Nancy's number, we will solve the equation \( 6x + 9 = 75 \). 1. Subtract 9 from both sides: \[ 6x = 75 - 9 \] \[ 6x = 66 \] 2. Divide both sides by 6: \[ x = \frac{66}{6} \] \[ x = 11 \] Thus, Nancy's number is \( 11 \). ### Summary of Answers (a) No, Nancy's number cannot be 9 because substituting 9 into the equation does not yield 75. (b) The techniques that could be used include: - A. Guess and check. - B. Eliminate possibilities. - C. Make an orderly list or table. - D. Use algebra. (c) Nancy's number is \( 11 \).