If perpendicular lines m and n intersect at (0,b) in the

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

Answer: C

Cheers,
Brent

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.

Answer: C

Can someone please explain in detail, how both are insufficient?

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.

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?

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

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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How did you get b = -2? Please show your work.
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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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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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Got my mistake, thanks Bunuel
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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

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)

Let equation of the two lines be

line n: y=nx+a ---1
line m: y=mx+c ---2

given they intersect at (0,b). Hence, (0,b) satisfies equation of both lines;
putting (0,b) in equation 1 we get b=a ; hence to find b we need to find value of a

statement one tells us m =-1/2 and therefore n = 2 as they are perpendicular to each other -- insufficient as it doesnt give us value of b

statement two tells us the point (-1,0) is on the line n
putting (-1,0) in eq 1 we get n=a
but we dont know the value of n

combining both we know n=2 and n=a=b therefore b =2

From (ii) (y-y1) = m (x-x1) ; where m is the slope of line.
substitute - y-0 = m(x+1)
=> y = mx + m {note: this m is nothing but y intercept }

from (i) we get the slope of line n, and that is 2.

clearly 1 and 2 independently are insuff.
together - put slope =2 in above equation we get the y-intercept =2 , and that is the value of 'b'

cheers!

Bunuel, GMAT Prep question. Please tag it.

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Done. Thank you!
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statement 1:
slope of m=-1/2 so slope of n=2
this does not help to find out about the value of b.
so insufficient

statement 2:
we cannot find value of b alone with this as well.

combining both statements:
we know slope of n=2 from statement1 and (-1,0) lie on line n
so equation of line n is
(y-y1)=n(x-x1)
y-0=2(x+1)
y=2x+2

we know (0,b) is also on line n, replacing the values in the equation we get
b=2(0) + 2
b=2
hence sufficient

Answer is C