Question

How do you find the slope and y intercept of $8x - 12y = 24$?

Answer 1

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 $8x - 12y = 24$.
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 8x - 12y = 24 \\\ \Rightarrow 12y = 8x - 24 \\\ \Rightarrow y = \dfrac{{8x}}{{12}} - \dfrac{{24}}{{12}} \\\ \Rightarrow y = \dfrac{{2x}}{3} - 2 \\\$
On comparing with the standard form of the slope-intercept formula, we see that
$m = \dfrac{2}{3}$ and $c = - 2$
Thus, the slope of the given line is $\dfrac{2}{3}$ and the y-intercept is $- 2$.
Hence, the slope of the line $8x - 12y = 24$ is $\dfrac{2}{3}$ and it cuts the y-axis at the point $(0,{\kern 1pt} {\kern 1pt} {\kern 1pt} - 2)$.

Thus, the slope of the given line is $\dfrac{2}{3}$ and the y-intercept is $- 2$.

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 as the value of $x$ increases and the value of $y$ decreases when the value of $x$ decreases. 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 $0$.