Question

How do you solve using the slope and intercept of the given linear equation, the equation is \[y = \dfrac{1}{3}x - 5\] ?

Answer 1

Hint: Slope is the angle from the positive axis of the graph to the line drawn; it gives the angle of inclination of the graph. To find the slope of any line you can use the trigonometric identity of “tanx” and can get the angle of slope by using it.

Complete step by step solution:
The given equation is \[y = \dfrac{1}{3}x - 5\]. According to the question we have to find the slope and intercept of the curve given, lets first find the slope of the graph: For finding slope we are using here the differentiation method of finding slope, in which we have to differentiate the equation as \[\dfrac{{dy}}{{dx}}\]
Now the value of “y” from the equation is:
\[y = \dfrac{1}{3}x - 5\]
Now on differentiating the term we get:
\[\dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\left( {\dfrac{1}{3}x - 5} \right) \\ \]
\[\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{1}{3} \]........(Differentiation of $x$ with respect to $x$ is $1$ and differentiation of constant terms is zero)
Here we obtain the slope of the given equation and it is a positive slope hence the equation lies in a positive slope. Now for finding the intercept we have to rearrange the equation as, the general equation which is:
\[ \Rightarrow \dfrac{x}{a} + \dfrac{y}{b} = 1\]..........(Here $a$ and $b$ are intercepts of the given equation)
Rearranging our equation we get:
$\Rightarrow \dfrac{x}{{3 \times ( - 5)}} + \dfrac{y}{{( - 5)}} = 1 \\$........(Dividing by $10$ on both sides of the equation)
\[\therefore - \dfrac{x}{{15}} + \left( { - \dfrac{y}{5}} \right) = 1 \\ \]

Hence, intercepts are $\dfrac{{ - 1}}{{15}},\dfrac{{ - 1}}{5}$ for $x$ and $y$ respectively.

Note: Here it was easy to rearrange the equation in the general form of the intercept equation but if this rearrangement is not possible then you have to plot the graph and the n check the points on which are curve is cutting the both axes and that coordinates are our required intercepts. The other method to solve the question is to first plot the graph then find the value from the graph.