To determine which ordered pair satisfies [tex]\( A \cap B \)[/tex], we need to find the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. This means we need to find the ordered pairs that satisfy both equations [tex]\( y = x \)[/tex] and [tex]\( y = 2x \)[/tex] simultaneously.
We will check each of the given ordered pairs one by one.
1. Ordered pair [tex]\((0, 0)\)[/tex]:
- For [tex]\( y = x \)[/tex]: [tex]\( 0 = 0 \)[/tex] is true.
- For [tex]\( y = 2x \)[/tex]: [tex]\( 0 = 2 \cdot 0 \)[/tex] which simplifies to [tex]\( 0 = 0 \)[/tex], and this is also true.
Hence, [tex]\((0, 0)\)[/tex] satisfies both equations.
2. Ordered pair [tex]\((1, 1)\)[/tex]:
- For [tex]\( y = x \)[/tex]: [tex]\( 1 = 1 \)[/tex] is true.
- For [tex]\( y = 2x \)[/tex]: [tex]\( 1 = 2 \cdot 1 \)[/tex] simplifies to [tex]\( 1 = 2 \)[/tex], which is false.
Hence, [tex]\((1, 1)\)[/tex] does not satisfy both equations.
3. Ordered pair [tex]\((1, 2)\)[/tex]:
- For [tex]\( y = x \)[/tex]: [tex]\( 2 = 1 \)[/tex] is false.
- For [tex]\( y = 2x \)[/tex]: [tex]\( 2 = 2 \cdot 1 \)[/tex] simplifies to [tex]\( 2 = 2 \)[/tex], which is true.
Hence, [tex]\((1, 2)\)[/tex] does not satisfy both equations.
4. Ordered pair [tex]\((2, 1)\)[/tex]:
- For [tex]\( y = x \)[/tex]: [tex]\( 1 = 2 \)[/tex] is false.
- For [tex]\( y = 2x \)[/tex]: [tex]\( 1 = 2 \cdot 2 \)[/tex] simplifies to [tex]\( 1 = 4 \)[/tex], which is false.
Hence, [tex]\((2, 1)\)[/tex] does not satisfy both equations.
After checking all the given ordered pairs, we conclude that the ordered pair [tex]\((0, 0)\)[/tex] is the only one that satisfies both [tex]\( y = x \)[/tex] and [tex]\( y = 2x \)[/tex].
Therefore, the ordered pair that satisfies [tex]\( A \cap B \)[/tex] is:
[tex]\[ (0, 0) \][/tex]
