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Question 8 of 10

Solve the system of equations:
[tex]\[
\begin{array}{l}
y = x - 3 \\
y = x^2 - 5x + 6
\end{array}
\][/tex]

A. [tex]\((2, 0)\)[/tex] and [tex]\((3, 0)\)[/tex]
B. [tex]\((1, -2)\)[/tex] and [tex]\((5, 2)\)[/tex]
C. [tex]\((0, -3)\)[/tex]
D. [tex]\((3, 0)\)[/tex]



Answer :

GinnyAnswer
To solve the given system of equations:
[tex]\[ \begin{array}{l} y = x - 3 \\ y = x^2 - 5x + 6 \end{array} \][/tex]

we follow these steps:

1. Set the equations equal to each other:
Since both equations are equal to [tex]\( y \)[/tex], we can set them equal:
[tex]\[ x - 3 = x^2 - 5x + 6 \][/tex]

2. Rearrange the equation:
Move all terms to one side to set the equation to zero:
[tex]\[ x^2 - 5x + 6 - x + 3 = 0 \][/tex]
Simplify:
[tex]\[ x^2 - 6x + 9 = 0 \][/tex]

3. Factor the quadratic equation:
Notice that this is a perfect square:
[tex]\[ (x - 3)^2 = 0 \][/tex]

4. Solve for [tex]\( x \)[/tex]:
Take the square root of both sides:
[tex]\[ x - 3 = 0 \][/tex]
[tex]\[ x = 3 \][/tex]

5. Find the corresponding [tex]\( y \)[/tex] value:
Use the original first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = x - 3 \][/tex]
Substitute [tex]\( x = 3 \)[/tex]:
[tex]\[ y = 3 - 3 = 0 \][/tex]

6. Write the solution pair:
The solution to the system of equations is:
[tex]\[ (3, 0) \][/tex]

Therefore, the correct answer is:
[tex]\[ \boxed{(3, 0)} \][/tex]

This corresponds to option D in the provided choices.

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