Question

What is the slope of the line \[3x + 2y + 1 = 0\]?
A) \[3/2\]
B) \[2/3\]
C) \[ - 3/2\]
D) \[ - 2/3\]

Answer 1

Hint:
We have a line represented by the equation\[ax + by + {c_0} = 0\], where \[a\], \[b\]and \[{c_0}\] are constants. In order to calculate the slope of that line, we convert it into another equation of the form \[y = mx + c\]. And it is the \['m'\]in this equation which is called the slope of that line.

Formula Used:
For the given question, we shall be using the formula for the slope of a line, which is:
\[m\](slope is represented by the small letter ‘m’) \[ = - \dfrac{{coefficient{\rm{ of x}}}}{{coefficient{\rm{ of y}}}}\] …(i)

Complete step by step solution:
If we have a line with an equation \[ax + by + {c_0} = 0\], where \[a\], \[b\] and \[{c_0}\] are constants, then in order to calculate the slope, we convert the equation in the form of \[y = mx + c\] …(ii)
And, \[m\] is the slope of the line.
Now, let us go back to the equation we had first, i.e., \[ax + by + {c_0} = 0\]
Keeping the \[y\] terms on the left-hand side of the equality and taking the other two terms,\[ax\]and \[{c_0}\], to the right-hand side, we get:
\[by = \left( { - a} \right)x + \left( { - {c_0}} \right)\]
Now, freeing the \[y\]on the left side of any coefficients by taking the coefficient to the right side of the equality, we get:
\[y = \left( { - \dfrac{a}{b}} \right)x + \left( { - \dfrac{{{c_0}}}{b}} \right)\]
Since both \[{c_0}\]and \[b\] are constants, \[\left( { - \dfrac{{{c_0}}}{b}} \right)\] is also a constant and can be replaced by another constant \[c\]
So, we have
\[y = \left( { - \dfrac{a}{b}} \right)x + c\]
Now, if we compare the above derived equation with the equation (ii), we obtain the value of the slope in terms of the coefficients of the line’s equation, so:
\[m = - \dfrac{a}{b}\]
Now, the equation given the question is:
\[3x + 2y + 1 = 0\]
Applying the formula for the slope:
here, we have \[a = 3\], \[b = 2\], so
Slope, \[m = - \dfrac{a}{b} = - \dfrac{3}{2}\]
\[\therefore m = - \dfrac{3}{2}\]

Hence, the correct option is C.

Note:
We saw that in these questions, if someone does not remember the formula for the slope of a line, they can easily derive it by themselves by just converting the equation of the line into the required form and equating the slope with the coefficients. The only thing to remember is that one needs to do the calculation very carefully as even the slightest mistake of a change of sign makes the answer wrong.