Consider the following sets.
[tex]$U =$[/tex] (ordered pairs on a coordinate plane)
[tex]$A =$[/tex] {ordered pair solutions to [tex]$y = x$[/tex]}
[tex]$B=\{$[/tex] ordered pair solutions to [tex]$y=2 x\}$[/tex]
Which ordered pair satisfies [tex]$A \cap B$[/tex]?
A. (0,0)
B. (1,1)
C. (1,2)
D. (2,1)
Answer (1)
The problem asks for the intersection of two sets, A and B , defined by the equations y = x and y = 2 x , respectively.
Test the ordered pair ( 0 , 0 ) : 0 = 0 and 0 = 2 ( 0 ) are both true.
Test the ordered pair ( 1 , 1 ) : 1 = 1 is true, but 1 = 2 ( 1 ) is false.
Test the ordered pair ( 1 , 2 ) : 2 = 1 is false, but 2 = 2 ( 1 ) is true.
Test the ordered pair ( 2 , 1 ) : 1 = 2 and 1 = 2 ( 2 ) are both false.
The ordered pair that satisfies both equations is ( 0 , 0 ) .
Explanation
Problem Analysis We are given two sets, A and B , defined by the equations y = x and y = 2 x , respectively. We want to find the ordered pair that satisfies both equations, which means we are looking for the intersection of the two sets, A c a pB . We will test each of the given ordered pairs to see if they satisfy both equations.
Testing (0,0) Let's test the ordered pair ( 0 , 0 ) .
For y = x , we have 0 = 0 , which is true. For y = 2 x , we have 0 = 2 ( 0 ) , which simplifies to 0 = 0 , which is also true. Since ( 0 , 0 ) satisfies both equations, it is a solution.
Testing (1,1) Let's test the ordered pair ( 1 , 1 ) .
For y = x , we have 1 = 1 , which is true. For y = 2 x , we have 1 = 2 ( 1 ) , which simplifies to 1 = 2 , which is false. Since ( 1 , 1 ) does not satisfy both equations, it is not a solution.
Testing (1,2) Let's test the ordered pair ( 1 , 2 ) .
For y = x , we have 2 = 1 , which is false. For y = 2 x , we have 2 = 2 ( 1 ) , which simplifies to 2 = 2 , which is true. Since ( 1 , 2 ) does not satisfy both equations, it is not a solution.
Testing (2,1) Let's test the ordered pair ( 2 , 1 ) .
For y = x , we have 1 = 2 , which is false. For y = 2 x , we have 1 = 2 ( 2 ) , which simplifies to 1 = 4 , which is also false. Since ( 2 , 1 ) does not satisfy both equations, it is not a solution.
Final Answer The only ordered pair that satisfies both equations y = x and y = 2 x is ( 0 , 0 ) . Therefore, the intersection of sets A and B is the ordered pair ( 0 , 0 ) .
Examples
In economics, the intersection of supply and demand curves represents the equilibrium point where the quantity supplied equals the quantity demanded. If y = x represents the supply curve and y = 2 x represents the demand curve, finding the intersection point helps determine the market equilibrium. In this case, the equilibrium point is ( 0 , 0 ) , indicating a scenario where there is no supply or demand at any positive price level. This concept is fundamental in understanding market dynamics and price determination.
