In the xy-plane, line l passes through the points (0,2) and (1,0), and

We can begin by finding an equation for each line:

Line l:

slope = (0-2)/(1-0) = -2, y-intercept = (0, 2) = 2

Thus, an equation for line l is y = -2x + 2

Line m:

Slope = (0-(-4)/(4-0) = 1, y-intercept = (0, -4) = -4

Thus, an equation for line m is y = x - 4.

Now, let’s find their intersection by setting the right hand side of the two equations equal to each other:

-2x + 2 = x - 4

-3x = -6

x = 2

Substitute x = 2 back to either equation (let’s take the one for line m), we have:

y = 2 - 4 = -2

We see that the point of intersection is (2, -2), so a = 2 > 0 and b = -2 < 0.

Alternate Solution:

Using the given points, make a quick sketch of the two lines on the same set of coordinate axes. Note that line l has a negative slope and line m has a positive slope. Their point of intersection will be in Quadrant 4, where x is always positive and y is always negative.

Answer: B
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The answer is B.

If you draw a line in XY plane, you can see they intersect in Quadrant-IV.

So, a>0 and b<0


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Thanks for the graphical solution. However, is there any algebraic approach as well that can be deployed to solve this question?

Graphical approach is recommended for this question as it is easy and short


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Hi Scott

Could you please elaborate on the alternate solution a bit more? A line that has a negative slope and a line that has a positive slope will always intersect in Quad 4? Not sure ive understood your alternate solution properly. Thanks in advance

Great question. By chance did you actually make a sketch of the given lines? If not, can you, and then let me know if the solution makes sense?
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Hey,

I tried making a negative sloping and a positive sloping line in every quadrant. I guess it can intersect in any quadrant you want it to.. but, basically if your x intercept is +ve then your y intercept will also be +ve and similarly for the opposite sign..

Given: In the xy-plane, line l passes through the points (0,2) and (1,0), and line m passes through the points (0,-4) and (4,0).

Asked: If (a,b) is the point of intersection of line l and line m, which of the following is true?

Equation of line l: -
y - 0 = (2-0)/(0-1) (x-1)
y = -2(x-1)

Equation of line m:-
y-0 = (-4-0)/(0-4) (x-4)
y = (x-4)

Point of Intersection of line l and line m:-
y = - 2(x-1) = x-4
-2x + 2 = x - 4
3x = 6
x = a = 2 > 0
y = b = -2 < 0


IMO B

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