To solve this problem, we need to identify the point(s) of intersection between the sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex].
1. Define the sets:
- Set [tex]\(A\)[/tex]: This set contains all the ordered pairs [tex]\((x, y)\)[/tex] such that [tex]\(y = x\)[/tex].
- Set [tex]\(B\)[/tex]: This set contains all the ordered pairs [tex]\((x, y)\)[/tex] such that [tex]\(y = 2x\)[/tex].
2. Intersection of sets [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
- [tex]\(A \cap B\)[/tex] includes the ordered pairs that satisfy both [tex]\(y = x\)[/tex] and [tex]\(y = 2x\)[/tex].
3. Find the common solutions:
- For a pair [tex]\((x, y)\)[/tex] to be in both [tex]\(A\)[/tex] and [tex]\(B\)[/tex], the following must be true simultaneously:
[tex]\[
y = x
\][/tex]
[tex]\[
y = 2x
\][/tex]
4. Solve the system of equations:
- Substitute [tex]\(y = x\)[/tex] into [tex]\(y = 2x\)[/tex]:
[tex]\[
x = 2x
\][/tex]
- Solve for [tex]\(x\)[/tex]:
[tex]\[
x - 2x = 0 \implies -x = 0
\implies x = 0
\][/tex]
5. Determine y:
- Substitute [tex]\(x = 0\)[/tex] back into [tex]\(y = x\)[/tex]:
[tex]\[
y = 0
\][/tex]
6. Identify the ordered pair:
- The ordered pair that satisfies both equations is [tex]\((0, 0)\)[/tex].
Therefore, the ordered pair that satisfies [tex]\(A \cap B\)[/tex] is [tex]\((0, 0)\)[/tex].
