To find the slope of the line that passes through the points \((-9, 41)\) and \((2, 3)\), we can use the formula for the slope \(m\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Here, \((x_1, y_1) = (-9, 41)\) and \((x_2, y_2) = (2, 3)\).
Plugging in the values:
\[ m = \frac{3 - 41}{2 - (-9)} = \frac{3 - 41}{2 + 9} = \frac{-38}{11} \]
So the slope is:
\[ \text{slope} = -\frac{38}{11} \]
Next, to find the vertical intercept, we can use the slope-intercept form of the line, which is:
\[ y = mx + b \]
We can use one of the given points to solve for \(b\) (the y-intercept). Let's use the point \((2, 3)\).
Substituting \(x = 2\), \(y = 3\), and \(m = -\frac{38}{11}\) into the equation:
\[ 3 = -\frac{38}{11}(2) + b \]
Calculating the value of \(-\frac{38}{11}(2)\):
\[ -\frac{76}{11} \]
Now substituting this into the equation:
\[ 3 = -\frac{76}{11} + b \]
To isolate \(b\), we can add \(\frac{76}{11}\) to both sides. First, convert 3 to a fraction with a denominator of 11:
\[ 3 = \frac{33}{11} \]
Now we can write:
\[ \frac{33}{11} + \frac{76}{11} = b \]
Combining the fractions:
\[ b = \frac{33 + 76}{11} = \frac{109}{11} \]
So the vertical intercept is:
\[ \text{vertical intercept} = \frac{109}{11} \]
Now we can write the equation of the line using the slope and the vertical intercept:
\[ y = mx + b \]
Substituting in the values we found:
\[ y = -\frac{38}{11}x + \frac{109}{11} \]
Thus, the equation of the line is:
\[ y = -\frac{38}{11}x + \frac{109}{11} \]
Summary:
- Slope: \(-\frac{38}{11}\)
- Vertical Intercept: \(\frac{109}{11}\)
- Equation of the Line: \(y = -\frac{38}{11}x + \frac{109}{11}\)
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