9. A linear function is given. \( \quad p(x)=23-8.6 x \) The function \( r(x) \) is also linear. The equation \( p(x)=r(x) \) has exactly one solution at \( x=5 \). Create two possible equations for the function \( r(x) \). \( r(x)= \) \( r(x)= \)

1. Since \( p(x) = 23 - 8.6x \) and the equation \( p(x) = r(x) \) has exactly one solution at \( x = 5 \), it must be that \[ r(5) = p(5). \] 2. Calculating \( p(5) \): \[ p(5) = 23 - 8.6 \times 5 = 23 - 43 = -20. \] Thus, we must have: \[ r(5) = -20. \] 3. Since \( r(x) \) is a linear function, we can write it in the form: \[ r(x) = mx + b. \] We need to choose values for \( m \) and \( b \) such that: \[ m \cdot 5 + b = -20. \] 4. One strategy is to choose an arbitrary value for \( m \) and then solve for \( b \). Two possible examples are: **Example 1: Let \( m = 1 \)** - Substituting into the equation: \[ 1 \cdot 5 + b = -20 \quad \Longrightarrow \quad 5 + b = -20 \quad \Longrightarrow \quad b = -25. \] - Thus, one possible function is: \[ r(x) = x - 25. \] **Example 2: Let \( m = 0 \)** - In this case, \( r(x) \) is a constant function: \[ 0 \cdot 5 + b = -20 \quad \Longrightarrow \quad b = -20. \] - Thus, another possible function is: \[ r(x) = -20. \] 5. Therefore, two possible equations for \( r(x) \) are: \[ r(x) = x - 25 \] \[ r(x) = -20. \]