Question

How do you write the slope-intercept form of the line $6x + 5y = - 15$?

Answer 1

Here, we are required to write the slope-intercept form of a line having the standard form $6x + 5y = - 15$. Thus, we will simply simplify this equation further such that we will be able to compare this to the general slope-intercept form and hence, it will be the required equation of the given line.

Formula Used:
General slope-intercept form of a line is given by: $y = mx + c$,
Where, $m$ is the slope and $c$is the $y$-intercept value.

Complete step by step solution:
According to the question, the given equation of the line is: $6x + 5y = - 15$
Clearly, this is in the standard form $Ax + By = C$
Thus, as we know, in order to find the slope-intercept form, we should know that the general slope-intercept form of a line is given by: $y = mx + c$
Where, $m$is the slope and $c$is the $y$-intercept value.
Thus, we will first of all subtract $6x$ from both the sides
Thus, we get,
$6x + 5y - 6x = - 15 - 6x$
$\Rightarrow 5y = - 6x - 15$
Now, dividing both sides by 5 in order to make the coefficient of $y$in the LHS as 1, we get,
$\Rightarrow \dfrac{{5y}}{5} = - \dfrac{6}{5}x - \dfrac{{15}}{5}$
$\Rightarrow y = - \dfrac{6}{5}x - 3$

Therefore, clearly, this is in the slope-intercept form $y = mx + c$
Hence, the slope-intercept form of the line $6x + 5y = - 15$ is $y = - \dfrac{6}{5}x - 3$
Thus, this is the required answer.

Note:
In geometry, a line can be defined as a straight one-dimensional figure that has no thickness and extends endlessly in both directions. It is sometimes described as the shortest distance between any two points. Whereas, a line segment is only a part of a line and it has two endpoints.
The standard form for linear equations in two variables or a line passing through two points is $Ax + By = C$. When an equation is given in this form then we can say that it is in the standard form, whereas, the slope-intercept form: $y = mx + c$ emphasizes on the slope, $m$ and the $y$-intercept of the line.