To solve the system of equations:
[tex]\[
\begin{array}{l}
y = x - 3 \\
y = x^2 - 5x + 6
\end{array}
\][/tex]
we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously.
### Step 1: Set the equations equal to each other
Since both equations are equal to [tex]\( y \)[/tex], we set them equal to each other:
[tex]\[ x - 3 = x^2 - 5x + 6 \][/tex]
### Step 2: Rearrange the equation
Subtract [tex]\( x - 3 \)[/tex] from both sides to get it into standard quadratic form:
[tex]\[ x - 3 - (x - 3) = x^2 - 5x + 6 - (x - 3) \][/tex]
[tex]\[ 0 = x^2 - 5x + 6 - x + 3 \][/tex]
[tex]\[ 0 = x^2 - 6x + 9 \][/tex]
### Step 3: Factor the quadratic equation
The quadratic equation [tex]\( x^2 - 6x + 9 \)[/tex] can be factored as:
[tex]\[ (x - 3)^2 = 0 \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Solve the factored equation:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]
### Step 5: Solve for [tex]\( y \)[/tex]
Substitute [tex]\( x = 3 \)[/tex] back into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = x - 3 \][/tex]
[tex]\[ y = 3 - 3 \][/tex]
[tex]\[ y = 0 \][/tex]
### Conclusion
The only solution to the system of equations is:
[tex]\[ (x, y) = (3, 0) \][/tex]
Thus, the correct answer is:
D. [tex]\((3, 0)\)[/tex]
