Question

Find the equation of the line having an inclination of $ {135^o} $ and the x-intercept of the line is $ 7 $ .

Answer 1

Hint: The inclination of the line is used to find the slope of the line while its x-intercept will give the point through which the line passes. The slope and the point should be substituted in the equation of the line passing through a point and have slope $ m $ .

Complete step-by-step answer:
Given information
Inclination of the line in degrees, $ \theta = {135^o} $ .
X-intercept of the line is, x=7 .
The inclination of the line will yield the slope of the line. As the slope is given by tangent of the angle.
Slope of the line,
 $ m = \tan \theta \cdots \left( 1 \right) $
Substitute the value of $ \theta = {135^o} $ in equation (1), we get
 $ \Rightarrow m = \tan {135^o} $
The value of $ \tan \left( {{{135}^o}} \right) = - 1 $ as the angle lies in the second quadrant. In the second quadrant, the value of the tangent of the angle is negative.
Therefore, $ m = - 1 $
The x-intercept of the line is given as $ 7 $ . It implies that the line intersects the X-axis at the point is $ \left( {7,0} \right) $ , it’s Y-coordinate is $ 0 $ .
Now, the equation of the line passing through the point $ \left( {{x_1},{y_1}} \right) $ and having slope is given by,
 $ \Rightarrow y - {y_1} = m\left( {x - {x_1}} \right) \cdots \left( 2 \right) $
Substitute the value of $ m = - 1 $ , $ {x_1} = 7 $ and $ {y_1} = 0 $ in equation (2), we get
 $ y - 0 = - 1\left( {x - 7} \right) $
Solving for $ y $ , we get
 $ y = 7 - x $
Thus, the equation of the line having an inclination of $ {135^o} $ and X-intercept $ 7 $ is $ y = 7 - x $ .

Note: The important concepts which should be kept in mind before solving such problems is that, The inclination of the line gives the slope of the line as the inclination is given in the form of angle and the tangent of the angle gives the slope of the line.
For instance, the line with an inclination of $ {45^o} $ has slope equal to $ m = \tan {45^o} = 1 $ . The intercept of the line on any axis gives the coordinates of the point through which a line passes. If the line intersects the x-axis then its Y-coordinate is $ 0 $ , while if it intersects the Y-axis then its X-coordinate is $ 0 $ .