Question

Write an equation of the slope-intercept form, of the line with an x-intercept at $(3,0)$ and a y-intercept at $(0, - 5)$
A. $y = \dfrac{4}{3}x - 3$
B. $y = \dfrac{5}{3}x - 5$
C. $y = 6x - 5$
D. $y = \dfrac{1}{2}x - 5$

Answer 1

Hint: Firstly, write the formula for the line equation in slope-intercept form and then substitute the value of y-intercept in that. Later substitute the value of x-intercept also to get the value of slope from the equation. Now again take a new slope-intercept equation and substitute the slope and then the y-intercept to get the required answer.

Formula used: Any straight line can be written in slope-intercept form, $y = mx + c$
Where $m$ is said to be the slope of the line $(m = \tan \theta )$
And $c$ is the y-intercept.

Complete step-by-step solution:
The given coordinates in the question are the x-intercept which is $(3,0)$ and the y-intercept $(0, - 5)$ Any straight line can be written in a slope-intercept form as $y = mx + c$
Given that the y-intercept is $(0, - 5)$
Here, $c = - 5$
Now substitute this in the slope-intercept form equation.
$ \Rightarrow y = mx - 5$
Since the x- intercept also lies on the line , it satisfies the equation
So, we substitute the x-intercept also to get the slope.
$ \Rightarrow 0 = m(3) - 5$
On simplifying further, we get slope as,
$ \Rightarrow 3m = 5$
$ \Rightarrow m = \dfrac{5}{3}$
Now since we have the slope and the y-intercept, write it directly into the slope-intercept formula to get the line equation we are evaluating for.
$ \Rightarrow y = mx + c$
On substituting the values,
$ \Rightarrow y = \dfrac{5}{3}x - 5$
$\therefore $ The line equation is $y = \dfrac{5}{3}x - 5$.

Option B is the correct answer.

Note: The slope of a line is the steepness of a line in a horizontal or vertical direction. The slope of a line can be calculated by taking the ratio of the change in vertical dimensions upon the change in horizontal dimensions.
This question can also be solved using the two-point line equation formula $\dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{x - {x_1}}}{{{x_2} - {x_1}}}$ . We must just by substituting the x-intercept and the y-intercept in the formula since both lie on the line equation and satisfy it.