To find the ordered pair [tex]\((x, y)\)[/tex] that satisfies the given system of equations, we have to check each of the possible pairs [tex]\((0, 2)\)[/tex], [tex]\((-4, 2)\)[/tex], [tex]\((2, 0)\)[/tex], and [tex]\((2, -4)\)[/tex] individually. Here's the detailed, step-by-step solution:
1. The given equations are:
[tex]\[
y = (x-2)(x+4)
\][/tex]
[tex]\[
y = 6x - 12
\][/tex]
2. Checking the pair (0, 2):
- For [tex]\(x = 0\)[/tex]:
[tex]\[
y = (0-2)(0+4) = (-2)(4) = -8
\][/tex]
[tex]\[
y = 6 \cdot 0 - 12 = -12
\][/tex]
This does not match [tex]\((0, 2)\)[/tex] since neither equation results in [tex]\(y = 2\)[/tex].
3. Checking the pair (-4, 2):
- For [tex]\(x = -4\)[/tex]:
[tex]\[
y = (-4-2)(-4+4) = (-6)(0) = 0
\][/tex]
[tex]\[
y = 6 \cdot -4 - 12 = -24 - 12 = -36
\][/tex]
This does not match [tex]\((-4, 2)\)[/tex] since neither equation results in [tex]\(y = 2\)[/tex].
4. Checking the pair (2, 0):
- For [tex]\(x = 2\)[/tex]:
[tex]\[
y = (2-2)(2+4) = (0)(6) = 0
\][/tex]
[tex]\[
y = 6 \cdot 2 - 12 = 12 - 12 = 0
\][/tex]
Both equations yield [tex]\(y = 0\)[/tex], so the pair (2, 0) is a solution to the system.
5. Checking the pair (2, -4):
- For [tex]\(x = 2\)[/tex]:
[tex]\[
y = (2-2)(2+4) = (0)(6) = 0
\][/tex]
[tex]\[
y = 6 \cdot 2 - 12 = 12 - 12 = 0
\][/tex]
This does not match [tex]\((2, -4)\)[/tex] since both equations result in [tex]\(y = 0\)[/tex], not [tex]\(-4\)[/tex].
After examining all the options, we find that the pair [tex]\((2, 0)\)[/tex] is the only pair that satisfies both equations. Therefore, the solution to the given system of equations is:
[tex]\[
\boxed{(2, 0)}
\][/tex]
