To find the initial value (y-intercept) \( b \) of the linear function, we can use the point-slope form of the linear equation given by:
\[ y = mx + b \]
Where:
- \( m \) is the slope,
- \( (x, y) \) is a point on the line,
- \( b \) is the y-intercept.
We are given:
- Rate of change (slope) \( m = -47 \)
- Point \( (x, y) = (14, 3) \)
Now, we can substitute these values into the equation to solve for \( b \):
\[ 3 = -47(14) + b \]
Calculating \( -47 \times 14 \):
\[ -47 \times 14 = -658 \]
Substituting this value into the equation:
\[ 3 = -658 + b \]
Now, isolate \( b \):
\[ b = 3 + 658 \] \[ b = 661 \]
The initial value (y-intercept) of the linear function is \( b = 661 \). However, since this option does not appear in the provided responses, let’s check if there was any mistake.
Rechecking Calculations:
- Rate of change \( m = -47 \)
- A point on the line: \( (14, 3) \)
Using \( y = mx + b \):
\[ 3 = -47 \times 14 + b \] Calculate \( -47 \times 14 \):
\[ -47 \times 14 = -658 \] So we have:
\[ 3 = -658 + b \] Thus:
\[ b = 3 + 658 = 661 \]
Conclusion: \( b = 661 \) and there must be a typo in the choices you provided. If you have to pick one from the options and they're incorrect, then kindly consult with your source or mentor. The correct value derived from the calculations is \( b = 661 \).