To address this question, we will graph the system of equations step by step and identify the points of intersection, which are the solutions to the system.
1. Graph the first equation [tex]\( y = -x + 2 \)[/tex]:
- This is a linear equation, representing a straight line.
- To graph it, we need to find at least two points on the line:
- Let [tex]\( x = 0 \)[/tex]: [tex]\( y = -0 + 2 = 2 \)[/tex]. This gives the point [tex]\((0, 2)\)[/tex].
- Let [tex]\( x = 2 \)[/tex]: [tex]\( y = -2 + 2 = 0 \)[/tex]. This gives the point [tex]\((2, 0)\)[/tex].
- Plot these points and draw a straight line through them.
2. Graph the second equation [tex]\( y = x^2 - 6x + 8 \)[/tex]:
- This is a quadratic equation, representing a parabola.
- To graph it, we should find the vertex and a few additional points:
- The vertex formula for the parabola [tex]\( y = ax^2 + bx + c \)[/tex] is given by [tex]\( x = -\frac{b}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -6 \)[/tex], so:
[tex]\[
x = -\frac{-6}{2 \times 1} = 3
\][/tex]
- Substitute [tex]\( x = 3 \)[/tex] back into the equation to find the vertex:
[tex]\[
y = (3)^2 - 6(3) + 8 = 9 - 18 + 8 = -1
\][/tex]
The vertex is at [tex]\((3, -1)\)[/tex].
- To find additional points, we can examine a few [tex]\( x \)[/tex]-values around the vertex:
- Let [tex]\( x = 0 \)[/tex]: [tex]\( y = (0)^2 - 6(0) + 8 = 8 \)[/tex]. This gives the point [tex]\((0, 8)\)[/tex].
- Let [tex]\( x = 1 \)[/tex]: [tex]\( y = (1)^2 - 6(1) + 8 = 1 - 6 + 8 = 3 \)[/tex]. This gives the point [tex]\((1, 3)\)[/tex].
- Let [tex]\( x = 2 \)[/tex]: [tex]\( y = (2)^2 - 6(2) + 8 = 4 - 12 + 8 = 0 \)[/tex]. This gives the point [tex]\((2, 0)\)[/tex].
- By plotting the vertex [tex]\((3, -1)\)[/tex] and a few additional points [tex]\((0, 8)\)[/tex], [tex]\((1, 3)\)[/tex], [tex]\((2, 0)\)[/tex], we can draw the parabolic curve.
3. Identify the intersections:
- Now that both the line and the parabola are graphed, we observe where they intersect. From the points discussed, we see the intersections at [tex]\((0, 2)\)[/tex] and [tex]\((2, 0)\)[/tex].
Thus, the solutions to the system of equations are the points where the line intersects the parabola. The intersections we have found are at [tex]\((0, 2)\)[/tex] and [tex]\((2, 0)\)[/tex].
Therefore, based on the given choices, the correct solution is:
A. Solutions: [tex]\( (0, 2) \)[/tex] and [tex]\( (2, 0) \)[/tex].
