To find the obtuse angle that the given line [tex]\( y - 2x = 7 \)[/tex] makes with the x-axis, let's follow these steps:
1. Rewrite the equation in slope-intercept form:
The given line equation is [tex]\( y - 2x = 7 \)[/tex]. We can rewrite it in the standard slope-intercept form [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
[tex]\[
y = 2x + 7
\][/tex]
Here, the slope [tex]\( m \)[/tex] is 2.
2. Determine the acute angle:
The angle that the line makes with the x-axis can be found using the arctangent function of the slope. This angle is the acute angle (since arctangent function always gives the smallest angle between the line and the x-axis).
3. Calculate the acute angle:
The acute angle [tex]\( \theta \)[/tex] is given by:
[tex]\[
\theta = \tan^{-1}(m) = \tan^{-1}(2)
\][/tex]
Converting this angle from radians to degrees, we find:
[tex]\[
\theta \approx 63.43^\circ
\][/tex]
4. Find the obtuse angle:
An obtuse angle is one that is greater than 90 degrees but less than 180 degrees. Since an acute angle is less than 90 degrees and the total angle in a straight line is 180 degrees, the obtuse angle [tex]\( \phi \)[/tex] can be found by subtracting the acute angle from 180 degrees.
[tex]\[
\phi = 180^\circ - \theta
\][/tex]
Substituting the value of [tex]\(\theta\)[/tex]:
[tex]\[
\phi = 180^\circ - 63.43^\circ = 116.57^\circ
\][/tex]
Thus, the obtuse angle that the line [tex]\( y - 2x = 7 \)[/tex] makes with the x-axis is approximately [tex]\( \boxed{116.57^\circ} \)[/tex].
