Question

How do you find the slope and intercept of $ - 3x - 5y + 35 = 0$ ?

Answer 1

Hint: In this question, we need to find the slope and intercept of the given straight line. Firstly, we will convert the given equation into a slope intercept form of a straight line. It can be done by first taking the constant term to the right hand side and then adding $3x$ on both sides of the given equation. Then dividing each term by 5 and rearranging the obtained equation. We know that the general equation of a straight line in slope intercept form is given by $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. Comparing the obtained equation with the general equation of a straight line we can determine the slope and intercept of the required line and obtain the result as wanted.

Complete step by step solution:
Given the equation $ - 3x - 5y + 35 = 0$ …… (1)
Note that the above expression given in the equation (1) is an equation of a straight line.
Here we are asked to find out the slope and intercept of the line given in the equation (1).
To find this, we need to convert our given equation into slope intercept form of a straight line.
The general equation of a straight line in slope intercept form is given by,
$y = mx + c$ …… (2)
where $m$ is the slope or gradient of a line and $c$ is the intercept of a line.
Now we convert the given equation of a line into slop intercept form by rearranging the terms.
Consider the equation of a line given in the equation (1).
Firstly, subtract the constant term which is 35 on both sides of the equation, we get,
$ \Rightarrow - 3x - 5y + 35 - 35 = 0 - 35$
$ \Rightarrow - 3x - 5y + 0 = 0 - 35$
$ \Rightarrow - 3x - 5y = - 35$
Adding $3x$ on both sides of the equation, we get,
$ \Rightarrow - 3x - 5y + 3x = - 35 + 3x$
Combining the like terms we get,
$ \Rightarrow - 3x + 3x - 5y = - 35 + 3x$
$ \Rightarrow 0 - 5y = - 35x + 3x$
$ \Rightarrow - 5y = - 35 + 3x$
Now dividing throughout by 5 we get,
$ \Rightarrow \dfrac{{ - 5y}}{5} = \dfrac{{ - 35 + 3x}}{5}$
$ \Rightarrow - y = \dfrac{{ - 35}}{5} + \dfrac{3}{5}x$
$ \Rightarrow - y = - 7 + \dfrac{3}{5}x$
Now multiplying by -1 on both sides, we get,
  $ \Rightarrow - y( - 1) = ( - 1)\left( { - 7 + \dfrac{3}{5}x} \right)$
Simplifying we get,
$ \Rightarrow y = 7 - \dfrac{3}{5}x$
Rearranging the terms in the R.H.S. we get,
$ \Rightarrow y = - \dfrac{3}{5}x + 7$ …… (3)
Comparing the equation (3) with the standard slope intercept form given in the equation (2), we get,
$m = - \dfrac{3}{5}$ and $c = 7$.
Hence we can conclude that the slope and intercept of the straight line$ - 3x - 5y + 35 = 0$ is given by $m = - \dfrac{3}{5}$ and $c = 7$.

Note: In this question, it is important to remember that $y = mx + c$ is the form called the slope intercept form of the equation of the straight line, where $m$ is the slope or gradient of a line and $c$ is the intercept of a line. It is the most popular form of the straight line.
Sometimes the given equation of a line won’t be the same as the general form. We need to convert or make rearrangement in such a way that the given expression becomes similar to the general slope intercept form. Then we compare them with each other and determine the values that are asked. Hence, it becomes easier to find the required things.