To determine which ordered pair satisfies \(A \cap B\), we first need to understand the definitions and properties of the sets provided and then find their intersection.
1. Define Set A and Set B:
- Set A: This set consists of all ordered pairs \((x, y)\) that satisfy the equation \(y = x\).
- Set B: This set consists of all ordered pairs \((x, y)\) that satisfy the equation \(y = 2x\).
2. Determine the Intersection \(A \cap B\):
- The intersection of sets \(A\) and \(B\), \(A \cap B\), includes all ordered pairs that satisfy both equations simultaneously.
3. Set the Equations Equal to Each Other:
- To find the common solution, we need to set the two equations equal to each other:
[tex]\[
y = x = 2x
\][/tex]
- Now, solve for \(x\):
[tex]\[
x = 2x
\][/tex]
- Subtract \(x\) from both sides:
[tex]\[
0 = x
\][/tex]
- Hence, \(x = 0\).
4. Determine the Corresponding Value of \(y\):
- Substitute \(x = 0\) back into either of the original equations to find \(y\):
[tex]\[
y = x = 0
\][/tex]
- Therefore, \(y = 0\).
5. Identify the Ordered Pair:
- The ordered pair that satisfies both equations simultaneously is \((0, 0)\).
6. Examine the Given Options:
- The options provided are:
- \((0, 0)\)
- \((1, 1)\)
- \((1, 2)\)
- \((2, 1)\)
- Out of these, only \((0, 0)\) satisfies both \(y = x\) and \(y = 2x\).
Therefore, the ordered pair that satisfies [tex]\(A \cap B\)[/tex] is [tex]\((0, 0)\)[/tex].
