To find the initial value (b) of the linear function, we can use the point-slope form of the equation of a line, which is given by:
\[ y = mx + b \]
where:
- \( y \) is the output (y-value),
- \( m \) is the slope (rate of change),
- \( x \) is the input (x-value),
- \( b \) is the y-intercept (initial value).
We are given:
- Slope \( m = -\frac{4}{7} \)
- A point \( (x, y) = (14, 3) \)
We can substitute \( m \), \( x \), and \( y \) into the equation to solve for \( b \):
\[ 3 = -\frac{4}{7}(14) + b \]
Now, calculate \( -\frac{4}{7}(14) \):
\[ -\frac{4}{7}(14) = -\frac{4 \times 14}{7} = -\frac{56}{7} = -8 \]
Now substitute this into the equation:
\[ 3 = -8 + b \]
To find \( b \), add 8 to both sides:
\[ b = 3 + 8 = 11 \]
Therefore, the initial value \( b = 11 \).
So the answer is:
b=11
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