Consider the following sets:

[tex]\[ U = \{\text{ordered pairs on a coordinate plane}\} \][/tex]
[tex]\[ A = \{\text{ordered pair solutions to } y = x \} \][/tex]
[tex]\[ B = \{\text{ordered pair solutions to } y = 2x \} \][/tex]

Which ordered pair satisfies [tex]\( A \cap B \)[/tex]?

A. [tex]\((0,0)\)[/tex]
B. [tex]\((1,1)\)[/tex]
C. [tex]\((1,2)\)[/tex]
D. [tex]\((2,1)\)[/tex]



Answer :

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].