Question

How do you find the equation of a line with slope of $- 3$ and contains the point $(0,\; - 3)?$

Answer 1

The equation of a line having slope equals to “m” and passing through a point having abscissa, which is the “x” value of the coordinate equals $x_1$ and ordinate, which is the “y” value of the coordinate equals $y_1$ is given as follows:
$\left( {y - y_1} \right) = m\left( {x - x_1} \right)$
Use this information to find the equation of the given line.

Formula used: Equation of a line when its slope and a passing point are given: $\left( {y - y_1} \right) = m\left( {x - x_1} \right)$ where $\left( {x_1,\;y_1} \right)$ is the coordinate of the passing point and $m$ is the slope of the line.

Complete step-by-step solution:
Let the point be “A” from which the given line having slope of $- 3$ is passing. Therefore the coordinates of point “A” will be given as ${\text{A}} \equiv \left( {{\text{0,}}\; - {\text{3}}} \right)$
Now we know that the equation of line having slope $m$ and passing through a point having coordinates $\left( {x_1,\;y_1} \right)$ is given as
$\left( {y - y_1} \right) = m\left( {x - x_1} \right)$
So we have given respective values of slope, $m = - 3$ and the coordinates $\left( {x_1,\;y_1} \right) = \left( {{\text{0,}}\; - {\text{3}}} \right)$ that is $x_1 = 0\;{\text{and}}\;y_1 = - 3$
Putting these values in the above equation in order to get the required equation of the line, we will get
$\Rightarrow \left( {y - \left( { - 3} \right)} \right) = - 3\left( {x - 0} \right)$
Simplifying this equation,
$\Rightarrow \left( {y + 3} \right) = - 3x \\\ \Rightarrow y + 3x + 3 = 0 \\\$
Therefore $y + 3x + 3 = 0$ is the required equation for the line passing through $(0,\; - 3)$ and having slope of $- 3$

Note: Slope of a line can be equals to zero or equals to infinity, according as the line is either parallel to x-axis or parallel to y-axis. Basically slope is the change of y values over x values we can also term it as “Rise over run”. Slope also has negative or positive values according to the nature of the line either decreasing or increasing.