Question

Find the equation of the line passing through points (5, 7) and (10, 11), then determine the angle of incline.

Answer 1

To find the equation of a line passing through the points \( (5, 7) \) and \( (10, 11) \), we first need to calculate the slope \( m \) of the line using the formula:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Here, \( (x_1, y_1) = (5, 7) \) and \( (x_2, y_2) = (10, 11) \). Plugging in these values:

\[
m = \frac{11 - 7}{10 - 5} = \frac{4}{5}
\]

Next, we can use the point-slope form of the equation of a line, which is given by:

\[
y - y_1 = m(x - x_1)
\]

Substituting \( m = \frac{4}{5} \) and the point \( (5, 7) \):

\[
y - 7 = \frac{4}{5}(x - 5)
\]

Now, simplify this equation:

\[
y - 7 = \frac{4}{5}x - \frac{4}{5} \cdot 5
\]
\[
y - 7 = \frac{4}{5}x - 4
\]
\[
y = \frac{4}{5}x - 4 + 7
\]
\[
y = \frac{4}{5}x + 3
\]

So the equation of the line is:

\[
y = \frac{4}{5}x + 3
\]

Next, we need to determine the angle of incline of the line. The angle \( \theta \) that the line makes with the positive x-axis can be calculated using the formula:

\[
\tan(\theta) = m
\]

Given that \( m = \frac{4}{5} \):

\[
\tan(\theta) = \frac{4}{5}
\]

To find \( \theta \), we take the arctangent:

\[
\theta = \arctan\left(\frac{4}{5}\right)
\]

Using a calculator:

\[
\theta \approx 38.66^\circ
\]

Thus, the equation of the line is \( y = \frac{4}{5}x + 3 \) and the angle of incline is approximately \( 38.66^\circ \).