To determine the solution set of the system of equations:
[tex]\[ y = x^2 - 6x + 12 \][/tex]
[tex]\[ y = 2x - 4 \][/tex]
we need to find the points [tex]\((x, y)\)[/tex] where both equations intersect. Let's follow the steps to determine the solution(s):
### Step 1: Set the equations equal to each other.
Given the equations:
[tex]\[ y = x^2 - 6x + 12 \][/tex]
[tex]\[ y = 2x - 4 \][/tex]
Setting them equal to find where they intersect:
[tex]\[ x^2 - 6x + 12 = 2x - 4 \][/tex]
This step is shown in the table as follows:
[tex]\[ x^2 - 6x + 12 = 2x - 4 \][/tex]
### Step 2: Simplify the equation.
We need to move all terms to one side of the equation to set it equal to zero:
[tex]\[ x^2 - 6x + 12 - 2x + 4 = 0 \][/tex]
Simplify the equation:
[tex]\[ x^2 - 8x + 16 = 0 \][/tex]
This simplified equation is shown in the table as:
[tex]\[ x^2 - 8x + 16 = 0 \][/tex]
### Step 3: Solve the quadratic equation.
The equation [tex]\(x^2 - 8x + 16 = 0\)[/tex] can be solved by factoring or using the quadratic formula. Here, it's straightforward to recognize that it is a perfect square:
[tex]\[ (x - 4)^2 = 0 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ x - 4 = 0 \][/tex]
[tex]\[ x = 4 \][/tex]
So, we have one solution: [tex]\(x = 4\)[/tex].
### Step 4: Find the corresponding [tex]\(y\)[/tex] value.
To find the [tex]\(y\)[/tex] value corresponding to [tex]\(x = 4\)[/tex], we can substitute [tex]\(x = 4\)[/tex] back into either original equation. Let's use the second equation [tex]\(y = 2x - 4\)[/tex]:
If [tex]\(x = 4\)[/tex]:
[tex]\[ y = 2(4) - 4 \][/tex]
[tex]\[ y = 8 - 4 \][/tex]
[tex]\[ y = 4 \][/tex]
So, the corresponding point is [tex]\((4, 4)\)[/tex].
### Final answer:
The point of intersection, and thus the solution to the system of equations, is:
[tex]\[ (4, 4) \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{(4,4)} \][/tex]
