Question

How do you find a standard form of equation of the line with slope of a line is $\dfrac{-1}{3}$ and y intercept is $\dfrac{10}{3}$?

Answer 1

Now we are given with the slope and y intercept of the line. We know the equation of line in slope intercept form is $y=mx+c$ where m is the slope of line and c is the y intercept of the line. Hence we can easily write the equation of line. Now we will simplify the equation and write it in the form $ax+by+c=0$

Complete step by step solution:
Now let us know that the equation of a line is a linear equation in two variables of the form $ax+by+c=0$ .
Now we are given the slope and y intercept of the line.
Slope of the line is the ratio $\dfrac{y}{x}$ at any point on the line.
Now y intercept of the line is the value of y when x is substituted as 0. Hence y intercept is nothing but the intersection of line and y axis.
Now we know that if m is the slope of the line and c is y intercept then the equation of the line in slope intercept form is given by $y=mx+c$.
Now we are given the slope of line is $\dfrac{-1}{3}$ and y intercept is $\dfrac{10}{3}$ . Hence the equation of the line is $y=\dfrac{-x}{3}+\dfrac{10}{3}$ .
Now the general form of the equation is $ax+by+c=0$ hence we will write the equation in this form.
Now consider $y=\dfrac{-x}{3}+\dfrac{10}{3}$ . Multiplying the whole equation by 3 we get,
$\Rightarrow 3y=-x+10$
Now rearranging the terms we get,
$\Rightarrow x+3y-10=0$

Hence the equation of the line in general form is $x+3y-10=0$.

Note: Now note that the slope of the line is the ratio of y and x on the line. Hence it is constant throughout the line. Now the slope is nothing but the value of $\tan \theta$ where $\theta$ is the angle made by the line on the x axis.