Bunuel wrote:
If \(ac \ne 0\), do graphs \(y=ax^2+b\) and \(y=cx^2+d\) intersect?
(1) \(a = -c\)
(2) \(b \gt d\)
Target question: Do the graphs y=ax²+b and y= cx²+d intersect?This is a good candidate for
rephrasing the target question. Key concept: If two lines (or curves) intersect at a point, then the COORDINATES of that intersection point must satisfy BOTH equations.
Since BOTH of the given equations are set equal to y, we can look for a value of x that yields the same value of y
In other words, if there's a solution to the equation ax²+b = cx²+d, then the two graphs intersect.
REPHRASED target question: Is there a value of x that satisfies the equation ax²+b = cx²+ d?Head straight to . . .
Statements 1 and 2 combined There are several values of a, b, c and d that satisfy BOTH statements. Here are two:
Case a: a = -1, c = 1, b = 4 and d = 2. Our equation becomes (-1)x² + 4 = (1)x²+ 2
Rearrange to get: 2 = 2x²
One solution is x = 1
In this case, the answer to the REPHRASED target question is
YES, there is a value of x that satisfies the equation ax²+b = cx²+ dCase b: a = 1, c = -1, b = 4 and d = 2. Our equation becomes (1)x² + 4 = (-1)x²+ 2
Rearrange to get: 2x² = -2
This means x² = -1, and there is no solution to this equation
In this case, the answer to the REPHRASED target question is
NO, there is no value of x that satisfies the equation ax²+b = cx²+ dSince we cannot answer the
target question with certainty, the combined statements are NOT SUFFICIENT
Answer: E
Cheers,
Brent
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