Answer :
To solve the problem of identifying which system of equations has no solution, let's analyze each system individually:
### System 1:
[tex]\[ \begin{cases} y = -3x + 8 \\ 6x + 2y = -4.5 \end{cases} \][/tex]
1. Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ 6x + 2(-3x + 8) = -4.5 \][/tex]
2. Simplify inside the parentheses:
[tex]\[ 6x - 6x + 16 = -4.5 \][/tex]
3. Combine like terms:
[tex]\[ 16 \neq -4.5 \][/tex]
This statement is false. Therefore, there is no value of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that can satisfy both equations simultaneously. Hence, System 1 has no solution.
### System 2:
[tex]\[ \begin{cases} y = 9x + 6.25 & y = 4.5x - 5 \\ -18x + 2y = 12.5 & -3x + 2y = 6 \end{cases} \][/tex]
1. Set the two [tex]\( y \)[/tex] equations equal to each other to find [tex]\( x \)[/tex]:
[tex]\[ 9x + 6.25 = 4.5x - 5 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ 9x - 4.5x = -5 - 6.25 \\ 4.5x = -11.25 \\ x = -2.5 \][/tex]
3. Substitute [tex]\( x \)[/tex] into either [tex]\( y \)[/tex] equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 9(-2.5) + 6.25 = -22.5 + 6.25 = -16.25 \][/tex]
Since we can find specific values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the equations, System 2 has solutions.
### System 3:
[tex]\[ \begin{cases} y = 3x + 9 \\ x + 8y = 12.3 \end{cases} \][/tex]
1. Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ x + 8(3x + 9) = 12.3 \][/tex]
2. Simplify inside the parentheses:
[tex]\[ x + 24x + 72 = 12.3 \][/tex]
3. Combine like terms and solve for [tex]\( x \)[/tex]:
[tex]\[ 25x + 72 = 12.3 \\ 25x = 12.3 - 72 \\ 25x = -59.7 \\ x = -2.388 \][/tex]
4. Substitute [tex]\( x \)[/tex] into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 3(-2.388) + 9 = -7.164 + 9 = 1.836 \][/tex]
Since we can find specific values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy these equations, System 3 has solutions.
### Conclusion
The system that has no solution is System 1:
[tex]\[ \begin{cases} y = -3x + 8 \\ 6x + 2y = -4.5 \end{cases} \][/tex]
### System 1:
[tex]\[ \begin{cases} y = -3x + 8 \\ 6x + 2y = -4.5 \end{cases} \][/tex]
1. Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ 6x + 2(-3x + 8) = -4.5 \][/tex]
2. Simplify inside the parentheses:
[tex]\[ 6x - 6x + 16 = -4.5 \][/tex]
3. Combine like terms:
[tex]\[ 16 \neq -4.5 \][/tex]
This statement is false. Therefore, there is no value of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that can satisfy both equations simultaneously. Hence, System 1 has no solution.
### System 2:
[tex]\[ \begin{cases} y = 9x + 6.25 & y = 4.5x - 5 \\ -18x + 2y = 12.5 & -3x + 2y = 6 \end{cases} \][/tex]
1. Set the two [tex]\( y \)[/tex] equations equal to each other to find [tex]\( x \)[/tex]:
[tex]\[ 9x + 6.25 = 4.5x - 5 \][/tex]
2. Solve for [tex]\( x \)[/tex]:
[tex]\[ 9x - 4.5x = -5 - 6.25 \\ 4.5x = -11.25 \\ x = -2.5 \][/tex]
3. Substitute [tex]\( x \)[/tex] into either [tex]\( y \)[/tex] equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 9(-2.5) + 6.25 = -22.5 + 6.25 = -16.25 \][/tex]
Since we can find specific values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the equations, System 2 has solutions.
### System 3:
[tex]\[ \begin{cases} y = 3x + 9 \\ x + 8y = 12.3 \end{cases} \][/tex]
1. Substitute [tex]\( y \)[/tex] from the first equation into the second equation:
[tex]\[ x + 8(3x + 9) = 12.3 \][/tex]
2. Simplify inside the parentheses:
[tex]\[ x + 24x + 72 = 12.3 \][/tex]
3. Combine like terms and solve for [tex]\( x \)[/tex]:
[tex]\[ 25x + 72 = 12.3 \\ 25x = 12.3 - 72 \\ 25x = -59.7 \\ x = -2.388 \][/tex]
4. Substitute [tex]\( x \)[/tex] into the first equation to find [tex]\( y \)[/tex]:
[tex]\[ y = 3(-2.388) + 9 = -7.164 + 9 = 1.836 \][/tex]
Since we can find specific values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy these equations, System 3 has solutions.
### Conclusion
The system that has no solution is System 1:
[tex]\[ \begin{cases} y = -3x + 8 \\ 6x + 2y = -4.5 \end{cases} \][/tex]
