How do you use the slope and y-intercept for $x + 8y = 0$?

Hint: In this question, we will write the given equation in terms of general slope-intercept form by moving x coordinate to the right side, and then by comparison we will find slope and y-intercept of the given line. The slope will be the coefficient of x and y-intercept will be the constant part.

Complete step-by-step solution:
In two-dimensional geometry, the equation of the line in slope-intercept form can be written as,
$y = mx + c$
Where x is the value of x-coordinate, and y is the value of y-coordinate.
Here, m represents the slope of the lines and c represents the y-intercept of the line.
In a line, the slope of the line is the value of the tangent function for the angle which a given line makes with the x-axis in the anticlockwise direction.
And, the y-intercept of a line is the point on the y-axis. The given line intersects the y axis.
Now, the given line in a question is,
$ \Rightarrow x + 8y = 0$
Now, subtract $x$ from both sides,
$ \Rightarrow x + 8y - x = 0 - x$
Simplify the terms,
$ \Rightarrow 8y = - x$
Divide both sides by 8,
$ \Rightarrow y = - \dfrac{1}{8}x$
So, the equation of the line $x + 8y = 0$ in slope-intercept form is $y = - \dfrac{1}{8}x$.
Now, for finding the value of slope and intercept of the given equation we equate above form a given line with standard slope form.
On equating both lines we have $m = - \dfrac{1}{8}$ which imply that the slope of the given line is $ - \dfrac{1}{8}$.
Also, the value of ‘c’ of a given line is 0.
This implies that the y-intercept of the given line is 0.

Hence, from above we see that the slope and y-intercept of the given line are $ - \dfrac{1}{8}$ and $0$ respectively.

Note: As, we know that there are different forms of line 1st is one point and slope form $\left\{ {y - {y_1} = m\left( {x - {x_1}} \right)} \right\}$, 2nd is two-point form $\left\{ {y - {y_1} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\left( {x - {x_1}} \right)} \right\}$ and 3rd is slope-intercept form $\left\{ {y = mx + c} \right\}$. So, one should be very careful regarding choosing the form of the line which is required in the given question to get slope and y-intercept.