What is the slope of line l that does not pass through the origin?

Question

What is the slope of line  l  that does not pass through the origin?
1) The  y -intercept of line  l  is half its  x -intercept.
2) The  y -intercept of line  l  is three units less than its  x -intercept.

yashikaaggarwal · Answer

1) The y-intercept of line l is half its x-intercept.
let x = 0 in y = mx + c
y = c
then X = 2C when Y = 0
0= mx+c
mx = -c
m(2C) = -c
2m = -1 
m = -1/2
slope is -1/2
(sufficient)

2) The y-intercept of line l is three units less than its x-intercept.
let x = 0 in y = mx + c
y = c
then X = C+3 when Y = 0
0= mx+c
mx = -c
m(C+3) = -c
mc + 3m = -c
c(m+1) = -3m
no value for m
(Insufficient)

IMO A

IanStewart · Answer

To find the x-intercept of a line y = mx + b, we replace y with zero and solve for x. So if we solve 0 = mx + b for x, the value of x we'll find is the x-intercept. b is the y-intercept, so if Statement 1 is true, b = 0.5x, and 0 = mx + 0.5x, and since the x-intercept is nonzero (the line does not pass through the origin), we can divide both sides by x to find m = -0.5, and Statement 1 is sufficient.

Statement 2 is not sufficient, as any numerical examples will show. If the x-intercept is 2, say, then the slope of the line will be positive, but if x = 5, say, then it will be negative, so Statement 2 cannot be sufficient.

Statement 2 is not sufficient, as any numerical examples will show. If the x-intercept is 2, say, then the slope of the line will be positive, but if x = 5, say, then it will be negative, so Statement 2 cannot be sufficient.

Lipun · Answer

Hi  IanStewart  sir,

Can you please explain as to why are we not considering the absolute values of the intercepts while evaluating statement 1? If we consider absolute value of the intercepts, then the slope can be 1/2 or -1/2.

Let me know if my understanding is incorrect.

Regards
Lipun

IanStewart · Answer

I'm not sure I understand your question - why is it that you think we need to use absolute value here? Intercepts can be positive or negative. If Statement 1 is true, then one intercept is half of the other, so they must have the same sign. If a line's x-intercept and y-intercept have the same sign, the line's slope is automatically negative, so the slope cannot be 1/2.

Mo2men · Answer

Hi  IanStewart

Actually I had same question. According to statement 1, y intercept is half x intercept. For example, if y =1, then x=2 or y=-1 then x =-2 but why does it have to be same sign. We still could have y =1 and x=-2. I solved it conceptually because nothing in statement 1 indicates we have same sign. Where did I go wrong?

Thanks

firas92 · Answer

Hi  Lipun   Mo2men

I don't understand why absolute values need to come into the picture.

Consider this, if the x-intercept of the line is  a , then the y intercept must be  \frac{a}{2}

So the line passes through the points  (a,0)  and  (0,\frac{a}{2})

We have a formula to calculate the slope of a line passing through the points  (x1,y1)  and  (x2,y2)

Slope=\frac{y2-y1}{x2-x1}

Using this formula,

Slope of line  l  =  \frac{\frac{a}{2}-0}{0-a}=-\frac{1}{2}

Hope this is clear

Posted from my mobile device

IanStewart · Answer

The solution firas92 posted above is correct. You can't have y = 1 and x = -2, because 1 is not half of -2; -1 is half of -2.

I'm wondering if perhaps you and Lipun are interpreting Statement 1 to mean "the distance from the y-intercept to the origin is half of the distance from the x-intercept to the origin," or something similar. If Statement 1 did say that, then you would both be correct. But that's not what it says - it is describing the intercepts themselves, and intercepts are coordinates, not distances, so they can be positive or negative. If you divide an intercept by 2, the sign won't change.

What is the slope of line l that does not pass through the origin?

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What is the slope of line l that does not pass through the origin?

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