To solve the system of equations:
[tex]\[
\begin{aligned}
&1. \quad 7x - y = 7 \\
&2. \quad 7x^2 - y = 7
\end{aligned}
\][/tex]
we need to find values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations simultaneously.
Step-by-Step Solution:
1. Rearrange the first equation to express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[
7x - y = 7 \implies y = 7x - 7
\][/tex]
2. Substitute [tex]\(y = 7x - 7\)[/tex] into the second equation:
[tex]\[
7x^2 - (7x - 7) = 7
\][/tex]
3. Simplify the substituted equation:
[tex]\[
7x^2 - 7x + 7 = 7
\][/tex]
4. Subtract 7 from both sides to simplify further:
[tex]\[
7x^2 - 7x = 0
\][/tex]
5. Factor out the common term [tex]\(7x\)[/tex]:
[tex]\[
7x(x - 1) = 0
\][/tex]
6. Solve for [tex]\(x\)[/tex]:
[tex]\[
7x = 0 \quad \text{or} \quad x - 1 = 0 \\
x = 0 \quad \text{or} \quad x = 1
\][/tex]
7. Find the corresponding [tex]\(y\)[/tex] values for each [tex]\(x\)[/tex] value by substituting into [tex]\(y = 7x - 7\)[/tex]:
- For [tex]\(x = 0\)[/tex]:
[tex]\[
y = 7(0) - 7 = -7
\][/tex]
So, one solution is [tex]\((0, -7)\)[/tex].
- For [tex]\(x = 1\)[/tex]:
[tex]\[
y = 7(1) - 7 = 0
\][/tex]
So, another solution is [tex]\((1, 0)\)[/tex].
Conclusion:
The solutions to the system of equations are [tex]\((0, -7)\)[/tex] and [tex]\((1, 0)\)[/tex].
Checking the provided options:
- [tex]\((0, 7)\)[/tex] and [tex]\((0, -7)\)[/tex]: The second pair is correct but [tex]\( (0, 7) \)[/tex] is not a valid solution.
- [tex]\((1, 0)\)[/tex] and [tex]\((0, -7)\)[/tex]: Both pairs are correct.
- [tex]\((3, 14)\)[/tex] and [tex]\((0, -7)\)[/tex]: Only [tex]\( (0, -7) \)[/tex] is correct.
- [tex]\((4, 14)\)[/tex] and [tex]\((-1, 10)\)[/tex]: Neither pair is correct.
Hence, the correct answer is:
[tex]\[
\boxed{(1,0) \text{ and } (0,-7)}
\][/tex]
