gmatt1476 wrote:
In the standard (x,y) coordinate plane, what is the slope of the line containing the distinct points P and Q ?
(1) Both P and Q lie on the graph of |x| + |y| = 1.
(2) Both P and Q lie on the graph of |x + y| = 1.
Target question: What is the slope of the line containing the distinct points P and Q ? Statement 1: Both P and Q lie on the graph of |x| + |y| = 1 Since most people aren't familiar with the graph of |x| + |y| = 1, it makes sense to
test some valuesThere are infinitely many values of x and y that satisfy statement 1. Here are two:
Case a: For point P, x = 1 and y = 0 (1,0) and, for point Q, x = 0 and y = 1 (0,1). In this case,
the slope of the line = (1 - 0)/(0 - 1) = -1Case b: For point P, x = -1 and y = 0 (-1,0) and, for point Q, x = 0 and y = 1 (0,1). In this case,
the slope of the line = (1 - 0)/(0 - -1) = 1Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: Both P and Q lie on the graph of |x + y| = 1Before we do anything else, let's first check to see whether we can
re-test the same values we used to show that statement 1 is not sufficient.
Yes, it turns out that we CAN re-use those same values to get:
Case a: For point P, x = 1 and y = 0 (1,0) and, for point Q, x = 0 and y = 1 (0,1). In this case,
the slope of the line = (1 - 0)/(0 - 1) = -1Case b: For point P, x = -1 and y = 0 (-1,0) and, for point Q, x = 0 and y = 1 (0,1). In this case,
the slope of the line = (1 - 0)/(0 - -1) = 1Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined Since we were able to use the
same counter-examples to show that each statement ALONE is not sufficient, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: For point P, x = 1 and y = 0 (1,0) and, for point Q, x = 0 and y = 1 (0,1). In this case,
the slope of the line = (1 - 0)/(0 - 1) = -1Case b: For point P, x = -1 and y = 0 (-1,0) and, for point Q, x = 0 and y = 1 (0,1). In this case,
the slope of the line = (1 - 0)/(0 - -1) = 1Since we cannot answer the
target question with certainty, the combined statements are NOT SUFFICIENT
Answer: E
Cheers,
Brent
Hi, Brent, thank you for the precise explanation.
I think I’ve seen somewhere that when we see an equation such as this, \(|x|+|y|=1\), we can simply calculate in this way: \(x+y=1\), because anyway those are positives.
So I thought that I can make the equation \(y=-x+1\). Thus the slope is \(-1\).
What is wrong with thinking this way and do you know in what kind of situation we should consider \(|x|+|y|=1\) \(x+y=1\)?