To determine the ordered pair that satisfies the intersection of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we need to find a point that lies on both lines, [tex]\( y = x \)[/tex] and [tex]\( y = 2x \)[/tex].
1. Set Equations Equal:
We need [tex]\( y \)[/tex] to satisfy both equations simultaneously.
[tex]\[
y = x \quad \text{(1)}
\][/tex]
[tex]\[
y = 2x \quad \text{(2)}
\][/tex]
2. Equate the Equations:
Since both expressions represent [tex]\( y \)[/tex], we equate them:
[tex]\[
x = 2x
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
We solve for [tex]\( x \)[/tex] by isolating it in the equation:
[tex]\[
x - 2x = 0
\][/tex]
[tex]\[
-x = 0
\][/tex]
[tex]\[
x = 0
\][/tex]
4. Find Corresponding [tex]\( y \)[/tex]:
Substitute [tex]\( x = 0 \)[/tex] back into either of the original equations to find [tex]\( y \)[/tex]:
Using [tex]\( y = x \)[/tex]:
[tex]\[
y = 0
\][/tex]
Alternatively, using [tex]\( y = 2x \)[/tex] would yield the same result:
[tex]\[
y = 2 \cdot 0 = 0
\][/tex]
5. Conclusion:
Thus, the ordered pair [tex]\((0, 0)\)[/tex] is the point 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]. The only correct option from the choices given is:
[tex]\((0, 0)\)[/tex].
