Question

Find the slope of the line that passes through the given points. Then determine the Vertical Intercept. Use the Slope and Vertical Intercept to write the Equation for the Line. The first one is done for you as an example.

Note: If the slope does not exist, enter DNE
ordered pairs (-9, 41) and (2, 3)
slope
vertical intercept
equation of the line

Answer 1

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: \(y = -\frac{38}{11}x + \frac{109}{11}\)