Let's carefully analyze each system of equations to determine the number of solutions.
### System 1:
[tex]\[
\begin{cases}
-8x + 7y = -2 \\
8x - 7y = 2
\end{cases}
\][/tex]
To determine the number of solutions, we'll consider the possibilities:
- One solution: If the system is consistent and independent.
- No solution: If the system is inconsistent.
- Infinitely many solutions: If the system is consistent and dependent.
Here, the equations are:
[tex]\[
a_1 = -8, \, b_1 = 7, \, c_1 = -2
\][/tex]
[tex]\[
a_2 = 8, \, b_2 = -7, \, c_2 = 2
\][/tex]
By observing the system, the lines represented by these two equations intersect at exactly one point.
Thus, System 1 has 1 solution.
### System 2:
[tex]\[
\begin{cases}
8x - 7y = 2 \\
8x - 7y = 2
\end{cases}
\][/tex]
Both the equations are:
[tex]\[
a_1 = 8, \, b_1 = -7, \, c_1 = 2
\][/tex]
[tex]\[
a_2 = 8, \, b_2 = -7, \, c_2 = 2
\][/tex]
These equations are identical, meaning any (x, y) that satisfies one equation will satisfy the other.
Thus, System 2 has infinitely many solutions.
### System 3:
[tex]\[
\begin{cases}
8x + 7y = 30 \\
8x - 7y = 2
\end{cases}
\][/tex]
Here, the equations are:
[tex]\[
a_1 = 8, \, b_1 = 7, \, c_1 = 30
\][/tex]
[tex]\[
a_2 = 8, \, b_2 = -7, \, c_2 = 2
\][/tex]
Since the second equation is not a scalar multiple of the first, these two lines intersect at exactly one point.
Thus, System 3 has 1 solution.
### Summary:
Let's now map the results to their respective systems:
1. System 1:
[tex]\[
\begin{cases}
-8x + 7y = -2 \\
8x - 7y = 2
\end{cases}
\][/tex]
1 solution
2. System 2:
[tex]\[
\begin{cases}
8x - 7y = 2 \\
8x - 7y = 2
\end{cases}
\][/tex]
Infinitely many solutions
3. System 3:
[tex]\[
\begin{cases}
8x + 7y = 30 \\
8x - 7y = 2
\end{cases}
\][/tex]
1 solution
