Question

How do you find the slope of $9x - 6y + 18 = 0$?

Answer 1

The slope of a line in graph is the $\tan$ of the angle made by the line with the x-axis. In other words, it is the change in the value of $y$ with respect to $x$ in the equation. For a straight line, if two points $A({x_1},{\kern 1pt} {\kern 1pt} {\kern 1pt} {y_1})$ and $B({x_2},{\kern 1pt} {\kern 1pt} {\kern 1pt} {y_2})$ are situated on the line, then by slope formula we can calculate the slope (m) as, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$. Alternatively, we can also find the slope of the line by using the slope-intercept formula wherein we write the given equation in the form $y = mx + c$, where $m$ is the slope of the line and $c$ is the y-intercept.

Complete step by step solution:
We have to find the slope of the line given by the equation $9x - 6y + 18 = 0$.
We will use the slope-intercept formula to find the slope of the line.
The slope-intercept formula is given by $y = mx + c$.
We can rewrite the given equation in the form,
$\Rightarrow 9x - 6y + 18 = 0 \\\ \Rightarrow 6y = 9x + 18 \\\ \Rightarrow y = \dfrac{9}{6}x + \dfrac{{18}}{6} \\\ \Rightarrow y = \dfrac{3}{2}x + 3 \\\$
On comparing with the standard form of the slope-intercept formula, we see that
$m = \dfrac{3}{2}$ and $c = 3$
Thus, the slope of the given line is $\dfrac{3}{2}$ and the y-intercept is $3$.

Hence, the slope of the line $9x - 6y + 18 = 0$ is $\dfrac{3}{2}$.

Note: For a line making acute angle with the x-axis, the slope is positive as the behavior of $y$ is same as that of $x$, i.e. the value of $y$ increases for increase in the value of $x$ and the value of $y$ decreases for decrease in the value of $x$. Also, we will arrive at the same result if we use the slope formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ to find the slope.