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Solve the system of equations:

[tex]\[ \begin{array}{l} y = x - 3 \\ y = x^2 - 5x + 6 \end{array} \][/tex]

A. [tex]\((1, -2)\)[/tex] and [tex]\((5, 2)\)[/tex]

B. [tex]\((2, 0)\)[/tex] and [tex]\((3, 0)\)[/tex]

C. [tex]\((0, -3)\)[/tex]

D. [tex]\((3, 0)\)[/tex]



Answer :

GinnyAnswer
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]

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