Given the system of equations:
1. [tex]\((x-3)^2 = y - 4\)[/tex]
2. [tex]\(y = -x + b\)[/tex]
We need to determine the value of [tex]\(b\)[/tex] and identify the second solution to the system.
First, we'll determine the coordinates of the vertex of the parabola given by [tex]\((x-3)^2 = y - 4\)[/tex]. The vertex form of a parabola [tex]\((x-h)^2 = y - k\)[/tex] indicates the vertex is at [tex]\((h, k)\)[/tex]. Given [tex]\((x-3)^2 = y-4\)[/tex], the vertex [tex]\((h, k)\)[/tex] is at [tex]\((3, 4)\)[/tex].
This vertex must satisfy the linear equation [tex]\(y = -x + b\)[/tex]:
[tex]\[
4 = -3 + b
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
4 = -3 + b \implies b = 7
\][/tex]
So, [tex]\(b\)[/tex] must be [tex]\(7\)[/tex] for the vertex [tex]\((3, 4)\)[/tex] to be one of the solutions to the system.
Next, we'll check each of the given conditions to see if they meet the qualifications.
1. Condition: [tex]\(b = 7\)[/tex] and a second solution at the point [tex]\((2,5)\)[/tex].
- Substitute [tex]\(b = 7\)[/tex] in the linear equation: [tex]\(y = -x + 7\)[/tex].
- Check if [tex]\((2, 5)\)[/tex] satisfies the equations.
- For [tex]\((2, 5)\)[/tex]:
[tex]\[
y = -2 + 7 = 5
\][/tex]
- Check if [tex]\((2,5)\)[/tex] satisfies the parabola equation:
[tex]\[
(2-3)^2 = 5 - 4 \implies 1 = 1
\][/tex]
Hence, [tex]\((2, 5)\)[/tex] is a valid solution when [tex]\(b = 7\)[/tex].
2. Condition: [tex]\(b = 1\)[/tex] and a second solution at the point [tex]\((4, 5)\)[/tex].
- Substitute [tex]\(b = 1\)[/tex] in the linear equation: [tex]\(y = -x + 1\)[/tex].
- Check if [tex]\((4, 5)\)[/tex] satisfies the equations.
- For [tex]\((4, 5)\)[/tex]:
[tex]\[
y = -4 + 1 = -3
\][/tex]
Since [tex]\(-3 \neq 5\)[/tex], this condition fails.
3. Condition: [tex]\(b = 7\)[/tex] and a second solution at the point [tex]\((1, 8)\)[/tex].
- Substitute [tex]\(b = 7\)[/tex] in the linear equation: [tex]\(y = -x + 7\)[/tex].
- Check if [tex]\((1, 8)\)[/tex] satisfies the equations.
- For [tex]\((1, 8)\)[/tex]:
[tex]\[
y = -1 + 7 = 6
\][/tex]
Since [tex]\(6 \neq 8\)[/tex], this condition fails.
4. Condition: [tex]\(b = 1\)[/tex] and a second solution at the point [tex]\((3, 4)\)[/tex].
- Substitute [tex]\(b = 1\)[/tex] in the linear equation: [tex]\(y = -x + 1\)[/tex].
- Check if [tex]\((3, 4)\)[/tex] satisfies the equations.
- For [tex]\((3, 4)\)[/tex]:
[tex]\[
y = -3 + 1 = -2
\][/tex]
Since [tex]\(-2 \neq 4\)[/tex], this condition fails.
Hence, in order for Tom's thinking to be correct, the value [tex]\(b\)[/tex] must be [tex]\(7\)[/tex] and the second solution to the system must be located at the point [tex]\((2, 5)\)[/tex].
