Question

How do you find the slope and y-intercept of $3y = - 6x + 9$?

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 $2x - 5y = - 45$.
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 2x - 5y = - 45 \\
   \Rightarrow 2x + 45 = 5y \\
   \Rightarrow 5y = 2x + 45 \\
   \Rightarrow y = \dfrac{2}{5}x + \dfrac{{45}}{5} \\
   \Rightarrow y = \dfrac{2}{5}x + 9 \\
$
On comparing with the standard form of the slope-intercept formula, we see that
$m = \dfrac{2}{5}$ and $c = 9$
Thus, the slope of the given line is $\dfrac{2}{5}$ and the y-intercept is $9$.
Hence, the slope of the line $2x - 5y = - 45$ is $\dfrac{2}{5}$ and it cuts the y-axis at the point $(0,{\kern 1pt} {\kern 1pt} {\kern 1pt} 9)$.

Thus, the slope of the given line is $\dfrac{2}{5}$ and the y-intercept is $9$.

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 $0$.