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Determine the solution set of the system of equations [tex]\(y = x^2 - 6x + 12\)[/tex] and [tex]\(y = 2x - 4\)[/tex].

[tex]\[
\begin{array}{|c|c|}
\hline
\text{Step} & \text{Equation} \\
\hline
\text{Step 1} & x^2 - 6x + 12 = 2x - 4 \\
\hline
\text{Step 2} & x^2 - 8x + 16 = 0 \\
\hline
\end{array}
\][/tex]

Which represents the solution(s) of this system of equations?

A. [tex]\((4, 4)\)[/tex]
B. [tex]\((-4, -12)\)[/tex]
C. [tex]\((4, 4)\)[/tex] and [tex]\((-4, 12)\)[/tex]
D. [tex]\((-4, 4)\)[/tex] and [tex]\((4, 12)\)[/tex]

Sagot :

GinnyAnswer
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]
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