To find the initial value (b) of the linear function given the slope (m) and a specific point (x, y), we can use the slope-intercept form of the equation of a line:
\[ y = mx + b \]
We are given:
- The rate of change \( m = -\frac{4}{7} \)
- A point \( (14, 3) \)
Now, substitute the given values into the equation:
- Substitute \( y = 3 \), \( m = -\frac{4}{7} \), and \( x = 14 \):
\[ 3 = -\frac{4}{7}(14) + b \]
- Calculate \( -\frac{4}{7}(14) \):
\[ -\frac{4 \cdot 14}{7} = -\frac{56}{7} = -8 \]
- Now replace this into the equation:
\[ 3 = -8 + b \]
- Solve for \( b \):
\[ b = 3 + 8 \] \[ b = 11 \]
Thus, the initial value \( b \) is 11.
The correct option is:
- b = 11 (or b equals 11)