To solve the system of nonlinear equations, we need to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously. The given equations are:
1. [tex]\( y = x - 3 \)[/tex]
2. [tex]\( y = x^2 - 5x + 6 \)[/tex]
Step-by-Step Solution:
1. Set the equations equal to each other:
Since both expressions are equal to [tex]\(y\)[/tex], set the right-hand sides of the equations equal to each other:
[tex]\[
x - 3 = x^2 - 5x + 6
\][/tex]
2. Rearrange the equation:
Move all terms to one side of the equation to set it to zero:
[tex]\[
x^2 - 5x + 6 - x + 3 = 0
\][/tex]
Combine like terms:
[tex]\[
x^2 - 6x + 9 = 0
\][/tex]
3. Factor the quadratic equation:
The quadratic equation simplifies into:
[tex]\[
(x - 3)^2 = 0
\][/tex]
This implies that:
[tex]\[
x - 3 = 0 \implies x = 3
\][/tex]
4. Find the corresponding y-value:
Substitute [tex]\( x = 3 \)[/tex] back into the first equation [tex]\( y = x - 3 \)[/tex]:
[tex]\[
y = 3 - 3 \implies y = 0
\][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ (x, y) = (3, 0) \][/tex]
After cross-referencing with the options provided:
A. [tex]\((0, -3)\)[/tex]
B. [tex]\((1, -2)\)[/tex] and [tex]\((5, 2)\)[/tex]
C. [tex]\((3, 0)\)[/tex]
D. [tex]\((2, 0)\)[/tex] and [tex]\((3, 0)\)[/tex]
The correct answer is:
C. [tex]\((3, 0)\)[/tex]
