Bot GPT 4-o mini To find the value of \( b \) in the slope-intercept form of the equation of the line \( y = mx + b \), we need to first calculate the slope \( m \) of the line that passes through points \( M(4, 3) \) and \( N(7, 12) \).
The formula for the slope \( m \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Substituting in the coordinates of points \( M \) and \( N \):
- \( (x_1, y_1) = (4, 3) \)
- \( (x_2, y_2) = (7, 12) \)
Now, plug in the values:
\[ m = \frac{12 - 3}{7 - 4} = \frac{9}{3} = 3 \]
Now that we have the slope \( m = 3 \), we can use the point-slope form of the equation to find \( b \). We can plug one of the points into the slope-intercept form. Let's use point \( M(4, 3) \).
We substitute \( m = 3 \), \( x = 4 \), and \( y = 3 \) into the equation \( y = mx + b \):
\[ 3 = 3(4) + b \]
This simplifies to:
\[ 3 = 12 + b \]
To solve for \( b \):
\[ b = 3 - 12 = -9 \]
Therefore, the value of \( b \) is
\[ \boxed{-9} \]
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