Question

Which of the following lines has the same y-intercept as $Y = \dfrac{x}{2} - 3$ ?
$A.\;x + 2 = 3y \\\ B.\;y + 3 = x + 2 \\\ C.\;y + 3 = 2x \\\ D.\;\dfrac{y}{2} = x - 3 \\\$

Answer 1

First we have to find the y-intercept value for the given line $Y = \dfrac{x}{2} - 3$ by using slope-intercept form of the equation of line. Then using the same method, we have to find y-intercepts of all four lines given in four options. Obviously in this way, we will get the correct option.

Complete step-by-step answer:
The equation of line given here is:
$Y = \dfrac{X}{2} - 3……...….(1)$
The above equation is already in slope-intercept form of the equation of line.
As, we know that equation in slope-intercept form is given as
$Y = m X + c ……...….(2)$
Where m is the slope of line with x-axis and c is the y−intercept.
Comparing equation (1) and (2) , we have the values of m and c as follows:
m = $\dfrac{1}{2}$ and c = -3
Thus the y-intercept value of the given line is -3.
Now, we compute the y-intercept value for all the lines one by one.
First,

x + 2 = {\text{ }}3y \\\ \Rightarrow 3y = x + 2 \\\ \Rightarrow y = \dfrac{1}{3}x + \dfrac{2}{3} ……...….(3)$$ Thus comparing equation (3) with equation (1) we get, c= $\dfrac{2}{3}$ So, the y-intercept value of this line is $\dfrac{2}{3}$ . Second, $ y + 3 = x + 2 \\\ \Rightarrow y = x - 11...….(4)$ Thus comparing equation (4) with equation (1) we get, c= -11 So, the y-intercept value of this line is -11 . Third, $ y + 3 = 2x \\\ \Rightarrow y = 2x - 3 …..(5)$ Thus comparing equation (5) with equation (1) we get, c= -3 So, the y-intercept value of this line is -3 . Fourth, $ \dfrac{y}{2} = x - 3 \\\ \Rightarrow y = 2x - 6…..(6)$ Thus comparing equation (6) with equation (1) we get, c= - 6 So, the y-intercept value of this line is -6 . Thus, the equation $y+3=2x$ has the same y−intercept value as of the given line $Y = \dfrac{x}{2} - 3$. **So, the correct answer is “Option C”.** **Note:** In coordinate geometry the general standard equation of a line is ax + by + c = 0. The equation of line can be represented in different ways in different standard forms. These are: 1) Point-slope form, 2) Two-point form, 3) Slope-intercept form 4) Intercept form