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

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

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.

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

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

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

Find value of b.
1) We don’t know sign of b.
but still line equation of m can be written as y= -(1/2)x + b
and euation of n is y=2x + b …(logic slope of perpendiculars form a product of -1)
Plug in (0,b) in each case you get b=b. So Insuf.

2) I can find slope of n as b. (rise over run) My equation for n will be y=bx + b
Plug in -1,0 you get 0=0 Insuf.

Combine,
y=2x+b … plug in (-1,0)… you get b=2
C both are suff.