Question

consider the following sets.
u = {ordered pairs on a coordinate plane}
a = {ordered pair solutions to y = x}
b = {ordered pair solutions to y = 2x}
which ordered pair satisfies ( a cap b )?
( (0, 0) )
( (1, 1) )
( (1, 2) )
( (2, 1) )

Explanation:

Step1: Understand \( A \cap B \)

\( A \cap B \) means the ordered pairs that are in both set \( A \) (solutions to \( y = x \)) and set \( B \) (solutions to \( y = 2x \)). So we need to find ordered pairs that satisfy both equations \( y = x \) and \( y = 2x \).

Step2: Solve the system of equations

Set the two equations equal to each other: \( x = 2x \) (since \( y = x \) and \( y = 2x \), so \( x = 2x \)).
Subtract \( x \) from both sides: \( x - x = 2x - x \), which simplifies to \( 0 = x \).
Then substitute \( x = 0 \) into \( y = x \), we get \( y = 0 \). So the ordered pair is \( (0, 0) \). We can also check each option:

• For \( (0, 0) \): In \( A \): \( 0 = 0 \) (satisfies \( y = x \)); In \( B \): \( 0 = 2\times0 \) (satisfies \( y = 2x \)).
• For \( (1, 1) \): In \( B \): \( 1

eq2\times1 = 2 \) (does not satisfy \( y = 2x \)).

• For \( (1, 2) \): In \( A \): \( 2

eq1 \) (does not satisfy \( y = x \)).

• For \( (2, 1) \): In \( A \): \( 1

eq2 \) (does not satisfy \( y = x \)); In \( B \): \( 1
eq2\times2 = 4 \) (does not satisfy \( y = 2x \)).

Answer:

\( (0, 0) \)