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Manager  Joined: 08 Dec 2012
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If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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If perpendicular lines m and n intersect at (0,b) in the standard (x,y) plane, what is the value of b?

(1) The slope of the line m is -1/2

(2) The point (-1,0) is on the line n

Originally posted by nave on 19 Oct 2013, 13:12.
Last edited by Bunuel on 20 Oct 2013, 04:56, edited 1 time in total.
Renamed the topic and edited the question.
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nave wrote:
If perpendicular lines m and n intersect at (0,b) in the standard (x,y) plane, what is the value of b?

(1) The slope of the line m is -1/2

(2) The point (-1,0) is on the line n

Given: Perpendicular lines m and n intersect at (0,b)

Target question: What is the value of b?

Statement 1: The slope of the line m is -1/2
Since line n is PERPENDICULAR to line m, we know that line n has slope 2
However, we can raise and lower the two lines so that the value of b is different.
To see what I mean, here are two possible cases:  As you can see, with each case, we get a different value of b.
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The point (-1,0) is on the line n
So, we know that the lines are perpendicular AND line n goes through the point (-1, 0)
However, since we don't know the SLOPES of the two lines, the value of b can change.
To see what I mean, here are two possible cases:  As you can see, with each case, we get a different value of b.
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that line m has slope -1/2 and line n has slope 2
Statement 2 tells us that line n passes through (-1,0)
These two pieces of information, LOCK line n into just ONE POSSIBLE line
In fact, this also means the y-intercept (0, b) is also LOCKED in.
We get: This also means that line m is also LOCKED in. Since both lines are now locked in, there is only one possible value of b.
Are we going to bother to actually find the value of b?
No, that would be a waste of valuable time. We need only recognize that there is ONLY ONE possible pair of lines that satisfy the two statements, which means we COULD find the value of b
Since we COULD answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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nave81 wrote:
If perpendicular lines $$m$$ and $$n$$ intersect at $$(0,b)$$ in the standard (x,y) plane, what is the value of $$b$$?

1. The slope of the line $$m$$ is $$-\frac{1}{2}$$

2. The point $$(-1,0)$$ is on the line $$n$$

From F.S 1 , all we know is that the slope of the line n is 2. Clearly Insufficient.

From F.S 2, we know a point on the line n, but don't know the slope.Insufficient.

Taking both together, we know that slope of line $$n = \frac{(b-0)}{(0-(-1)}$$ = b. And given that the slope is 2. Thus, b=2. Sufficient.

C.
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To find b we need equation of line.
1. gives us equation of line m but we do not know x intercept so not sufficient.
2. we cannot find the equation of line n so not sufficient.

1&2 we know slope of m we can get slope of n (inverse negative) and can find equation of line n. which can give us b.

nave wrote:
If perpendicular lines m and n intersect at (0,b) in the standard (x,y) plane, what is the value of b?

(1) The slope of the line m is -1/2

(2) The point (-1,0) is on the line n

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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Can someone please explain in detail, how both are insufficient?
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If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If perpendicular lines m and n intersect at (0,b) in the standard (x,y) plane, what is the value of b?

(1) The slope of the line m is -1/2

(2) The point (-1,0) is on the line n

In the original condition, for the line, there are 2 variables(the slope and the standard y plane). Since there are 2 lines, there should be 4 variables. Also, the two lines perpendicularly meet and multiplication of the slope is -1, which makes 1 equation. In order to match with the number of euqations, you need 3 more equations. For 1) 1 equations, for 2) 1 equation, which is likely to make E the answer. In 1) & 2), the slope if line m is -1/2 and the slope of line n should be 2. So, b=2 is derived from (b-0/0-(-1))=2, which is unique and sufficient. Therefore, the answer is C.

 For cases where we need 3 more equations, such as original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 80% chance that E is the answer (especially about 90% of 2 by 2 questions where there are more than 3 variables), while C has 15% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since E is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, C or D.
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Could someone explain why st.2 is not sufficient? Wouldn't b = 0 from (-1,0)? I guess this is just for line n.
Mau5 - could you pleas explain why the slope of n would equal b?
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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nave wrote:
If perpendicular lines m and n intersect at (0,b) in the standard (x,y) plane, what is the value of b?

(1) The slope of the line m is -1/2

(2) The point (-1,0) is on the line n

1) slope of m is -1/2; however, there are infinite number of lines that can have slope of -1/2 and cross the y-axis at x=0. NS

2) (-1,0) is on line n. However, you don't know what is the slope of n; n can slope upwards or downwards to satisfy (0,b) since b can be negative or positive number.

1+2) slope of m is -1/2. Since m and n are perpendicular, that means n has slope of 2. Given n has the point (-1,0) with a slope of 2, you can find out where it crosses the y-axis. C
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Hey Bunuel,

In this question, how do we know "b" is positive? if "b" is negative then the value of b = -2

Hence I marked E. Can you explain why we didn't consider the case where "b" is negative
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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pikolo2510 wrote:
Hey Bunuel,

In this question, how do we know "b" is positive? if "b" is negative then the value of b = -2

Hence I marked E. Can you explain why we didn't consider the case where "b" is negative

How did you get b = -2? Please show your work.
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Bunuel wrote:
pikolo2510 wrote:
Hey Bunuel,

In this question, how do we know "b" is positive? if "b" is negative then the value of b = -2

Hence I marked E. Can you explain why we didn't consider the case where "b" is negative

How did you get b = -2? Please show your work.

Sure Bunuel Each statement is insufficient, so I am considering the case when we look at the both the statement together

Case 1: -
Equation of line M
y = (-1/2)*x + b

Equation of line N
y = 2*x + b

As (-1,0) lies on line N, it should satisfy Equation of line N and hence value of b comes to b=2

Case 2 : -
Equation of line M
y = (-1/2)*x - b

Equation of line N
y = 2*x - b

As (-1,0) lies on line N, it should satisfy Equation of line N and hence value of b comes to b=- 2

The question says the two lines intersect at (0, b) , I don't know if b is positive or negative, hence I considered two cases of b and got two values of b i.e. -2 and 2. Let me know your thoughts
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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pikolo2510 wrote:
Bunuel wrote:
pikolo2510 wrote:
Hey Bunuel,

In this question, how do we know "b" is positive? if "b" is negative then the value of b = -2

Hence I marked E. Can you explain why we didn't consider the case where "b" is negative

How did you get b = -2? Please show your work.

Sure Bunuel Each statement is insufficient, so I am considering the case when we look at the both the statement together

Case 1: -
Equation of line M
y = (-1/2)*x + b

Equation of line N
y = 2*x + b

As (-1,0) lies on line N, it should satisfy Equation of line N and hence value of b comes to b=2

Case 2 : -
Equation of line M
y = (-1/2)*x - b

Equation of line N
y = 2*x - b

As (-1,0) lies on line N, it should satisfy Equation of line N and hence value of b comes to b=- 2

The question says the two lines intersect at (0, b) , I don't know if b is positive or negative, hence I considered two cases of b and got two values of b i.e. -2 and 2. Let me know your thoughts

It's not y = mx - b. It's always $$y=mx+b$$, where: $$m$$ is the slope of the line and $$b$$ is the y-intercept of the line.
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Bunuel, can you pls explain how from statement 2 alone you can not assume b= -1? As points in perpendicular lines are opposite
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Re: If perpendicular lines m and n intersect at (0,b) in the  [#permalink]

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Simba9 wrote:
Bunuel, can you pls explain how from statement 2 alone you can not assume b= -1? As points in perpendicular lines are opposite

Hello

The lines intersect at point (0,b) while (-1,0) is just a point on line n. That deosnt mean the points (0,b) and (-1,0) will be same.
And also, the two points anyway cannot be same. Because point (0,b) means x coordinate of this point is 0 while for point (-1,0) x coordinate is -1. So there is no point in comparing these two points. (pun not intended) Re: If perpendicular lines m and n intersect at (0,b) in the   [#permalink] 18 Apr 2018, 22:06
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