Question

Find the equation of a line with slope is $- 1$ and cut off an intercept of $4$ units in the negative direction of the $y$-axis.

Answer 1

To solve this question use the point-intercept formula to find the equation of the line. The point $(x,y)$ on the line with slope $m$ and $y$ -intercept $c$ lies on the line if and only if $y = mx + c\;$.
$\therefore$ substitute $- 1$ as m and $- 4$ as $c$ into the equation $y = mx + c\;$ that is the equation of line.
Take $c$ as $-4$ as the intercept is made on the negative side of the $y$-axis.

Complete answer:
The slope of the line is given as $- 1$ and the intercept on the negative $y$ -axis is 4. Since the intercept is made on the negative side of the $y$ -axis therefore, the value of $c$ is $- 4$.
\Rightarrow $$$m = - 1$ \Rightarrow $$$c = - 4 The point-intercept formula to find the equation of the line is $$y = mx + c\;$$, wherem$is the slope and$c$is the$y$-intercept. Substitute$m = - 1$and$c = - 4$ into the equation of the line,
\Rightarrow $$$$y = mx + c\;
$\Rightarrow y = - 1x + ( - 4)$
$\Rightarrow y = - x - 4$
After rearrange the equation we get,
\Rightarrow $$$$y + x + 4 = 0

The equation of a line with slope is $- 1$ and cutting off an intercept of $4$ units in the negative direction of $y$-axis is $y + x + 4 = 0$.

Note:
If the intercept on the positive side of the $y$-axis then the value of $c$will be positive and If the intercept on the negative side of the $y$-axis then the value of $c$will be negative. In this question take intercept $c = - 4$.
Another method:
The line with the slope m cuts the y-axis from the negative direction at a distance c from the origin.
Here, $c$ represents the y-intercept of the line. That means $x$ -coordinate of the point is $0$ and the $y$-coordinate is $- c$ because the intercept on the negative side of the $y$-axis .The coordinates of the point where the line meet the y-axis is $(0, - c)$.
$\therefore$ The point is $(0, - 4)$ and slope is $- 1$ .
Therefore, by point-slope form with slope $m$and point $\left( {{x_{1,}}{y_1}} \right)$ to find the equation of line is,
$y - {y_1} = m(x - {x_1})$
Substitute ${x_1} = 0$,${y_1} = - 4$ and $m = - 1$ into the equation,
$\Rightarrow y - ( - 4) = ( - 1)(x - 0)$
$\Rightarrow y + 4 = - 1x$
$\Rightarrow y + 4 + x = 0$
The equation of a line with slope is $- 1$ and cutting off an intercept of $4$ units in the negative direction of $y$-axis is $y + x + 4 = 0$.