Let's analyze each pair of equations step by step to determine the number of solutions for each system.
### System 1:
Equations:
[tex]\[ \begin{cases}
x = y - 3 \\
2x - 2y = 6
\end{cases} \][/tex]
To solve this system:
1. Substitute [tex]\( x = y - 3 \)[/tex] into [tex]\( 2x - 2y = 6 \)[/tex].
2. This gives [tex]\( 2(y - 3) - 2y = 6 \)[/tex].
3. Simplify to get [tex]\( 2y - 6 - 2y = 6 \)[/tex].
4. This simplifies to [tex]\(-6 = 6\)[/tex].
This system is inconsistent because [tex]\(-6 \neq 6\)[/tex].
Result: no solution
### System 2:
Equations:
[tex]\[ \begin{cases}
5x + 2y = -7 \\
10x + 4y = -14
\end{cases} \][/tex]
To solve this system:
1. Note that [tex]\( 10x + 4y = -14 \)[/tex] is just a multiple of [tex]\( 5x + 2y = -7 \)[/tex].
2. This implies both equations represent the same line.
This system is dependent because both equations are multiples of each other.
Result: infinite number of solutions
### System 3:
Equations:
[tex]\[ \begin{cases}
y = 2x + 1 \\
4x - 2y = -2
\end{cases} \][/tex]
To solve this system:
1. Substitute [tex]\( y = 2x + 1 \)[/tex] into [tex]\( 4x - 2y = -2 \)[/tex].
2. This gives [tex]\( 4x - 2(2x + 1) = -2 \)[/tex].
3. Simplify to get [tex]\( 4x - 4x - 2 = -2 \)[/tex].
4. This simplifies to [tex]\(-2 = -2\)[/tex].
This implies a consistent system with one solution.
Result: one solution
### System 4:
Equations:
[tex]\[ \begin{cases}
9x + 6y = 15 \\
2y = -1 + 3x
\end{cases} \][/tex]
To solve this system:
1. Solve [tex]\( 2y = -1 + 3x \)[/tex] for [tex]\( y \)[/tex].
2. This gives [tex]\( y = \frac{3x - 1}{2} \)[/tex].
3. Substitute [tex]\( y = \frac{3x - 1}{2} \)[/tex] into [tex]\( 9x + 6y = 15 \)[/tex].
4. This gives [tex]\( 9x + 6(\frac{3x - 1}{2}) = 15 \)[/tex].
5. Simplify to get [tex]\( 9x + 9x - 3 = 15 \)[/tex].
6. Combine like terms to get [tex]\( 18x - 3 = 15 \)[/tex].
7. This simplifies to [tex]\( 18x = 18 \)[/tex], hence [tex]\( x = 1 \)[/tex].
8. Substitute [tex]\( x = 1 \)[/tex] into [tex]\( 2y = -1 + 3x \)[/tex] to find [tex]\( y \)[/tex].
9. This gives [tex]\( 2y = -1 + 3(1) = 2 \)[/tex].
10. Hence, [tex]\( y = 1 \)[/tex].
This is a consistent system with one unique solution.
Result: one solution
Therefore, the matching of systems to solutions is:
- [tex]\( \begin{cases} x = y - 3 \\ 2x - 2y = 6 \end{cases} \)[/tex] → no solution
- [tex]\( \begin{cases} 5x + 2y = -7 \\ 10x + 4y = -14 \end{cases} \)[/tex] → infinite number of solutions
- [tex]\( \begin{cases} y = 2x + 1 \\ 4x - 2y = -2 \end{cases} \)[/tex] → one solution
- [tex]\( \begin{cases} 9x + 6y = 15 \\ 2y = -1 + 3x \end{cases} \)[/tex] → one solution
