Question

How do you find the slope and y-intercept given $9x - 3y = - 1$?

Answer 1

Hint: The slope of a line in graph is the change in the value of $y$ with respect to $x$ in the equation, i.e. $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$. The y intercept is the point at which the line cuts the y-axis which we can find by putting $x = 0$. Alternatively, we can find the slope of the line and the y intercept simultaneously 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 and y intercept of the line given by the equation $9x - 3y = - 1$.
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 - 3y = - 1 \\
   \Rightarrow 3y = 9x + 1 \\
   \Rightarrow y = \dfrac{9}{3}x + \dfrac{1}{3} \\
   \Rightarrow y = 3x + \dfrac{1}{3} \\
$
On comparing with the standard form of the slope-intercept formula, we see that
$m = 3$ and $c = \dfrac{1}{3}$
Thus, the slope of the given line is $3$ and the y-intercept is $\dfrac{1}{3}$.
Hence, the slope of the line $9x - 3y = - 1$ is $3$ and it cuts the y-axis at the point $(0,{\kern 1pt} {\kern 1pt} {\kern 1pt} \dfrac{1}{3})$.

Thus, the slope of the given line is $3$ and the y-intercept is $\dfrac{1}{3}$.

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 $x$. We can also find the y intercept of the line by putting $x = 0$ in the equation as when the line is cutting the y-axis the value of $x$ is $0$.