To determine which of the given points [tex]\((-1, 7)\)[/tex], [tex]\((0, 3)\)[/tex], [tex]\((0, 7)\)[/tex], and [tex]\((1, 9)\)[/tex] satisfy the system of equations involving the quadratic function [tex]\(f(x) = 5x^2 + x + 3\)[/tex] and the linear function [tex]\(g(x)\)[/tex] with given values, we follow these steps:
1. Evaluate [tex]\( f(x) \)[/tex] at each [tex]\( x \)[/tex] value:
- For [tex]\((-1, 7)\)[/tex]:
[tex]\[
f(-1) = 5(-1)^2 + (-1) + 3 = 5(1) - 1 + 3 = 5 - 1 + 3 = 7
\][/tex]
- For [tex]\((0, 3)\)[/tex]:
[tex]\[
f(0) = 5(0)^2 + 0 + 3 = 0 + 0 + 3 = 3
\][/tex]
- For [tex]\((0, 7)\)[/tex]:
Since [tex]\( f(0) = 3 \)[/tex] (from previous calculation), [tex]\( (0, 7) \)[/tex] does not satisfy [tex]\( f(x) \)[/tex].
- For [tex]\((1, 9)\)[/tex]:
[tex]\[
f(1) = 5(1)^2 + 1 + 3 = 5(1) + 1 + 3 = 5 + 1 + 3 = 9
\][/tex]
2. Check if [tex]\( y \)[/tex] in each point matches [tex]\( f(x) \)[/tex] and if these points are included in the given table of [tex]\( g(x) \)[/tex]:
- For [tex]\((-1, 7)\)[/tex]:
- [tex]\(f(-1) = 7 \)[/tex]
- Is [tex]\( (x, y) = (-1, 7) \)[/tex] in the table of [tex]\(g(x)\)[/tex]?
- The [tex]\( g(x) \)[/tex] values corresponding to [tex]\( x = -1 \)[/tex] is 5. So, this point does not satisfy [tex]\(g(x)\)[/tex].
- For [tex]\((0, 3)\)[/tex]:
- [tex]\(f(0) = 3 \)[/tex]
- Is [tex]\( (x, y) = (0, 3) \)[/tex] in the table of [tex]\(g(x)\)[/tex]?
- The [tex]\( g(x) \)[/tex] values corresponding to [tex]\( x = 0 \)[/tex] is 7. So, this point does not satisfy [tex]\(g(x)\)[/tex].
- For [tex]\((0, 7)\)[/tex]:
- [tex]\(f(0) = 3 \)[/tex]
- Since [tex]\( y = 7 \)[/tex], this point does not satisfy [tex]\(f(x)\)[/tex].
- For [tex]\((1, 9)\)[/tex]:
- [tex]\(f(1) = 9 \)[/tex]
- Is [tex]\( (x, y) = (1, 9) \)[/tex] in the table of [tex]\(g(x)\)[/tex]?
- The [tex]\( g(x) \)[/tex] values corresponding to [tex]\( x = 1 \)[/tex] is 9. So, this point satisfies both [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
3. Conclusion:
The points [tex]\((-1, 7)\)[/tex], [tex]\((0, 3)\)[/tex], and [tex]\((1, 9)\)[/tex] are the solutions that satisfy the system of equations involving both [tex]\(f(x)\)[/tex] and [tex]\(g(x)\)[/tex].
