Let's classify each of the three given systems of linear equations step by step.
### First System
[tex]\[
\begin{array}{l}
x + 3y = 4 \\
3x + 9y = 12
\end{array}
\][/tex]
To classify this system, notice the relationships between the equations. If we multiply the first equation by 3:
[tex]\[
3(x + 3y) = 3 \cdot 4 \\
3x + 9y = 12
\][/tex]
This is the exact same as the second equation. Therefore, these two equations are not linearly independent, meaning they actually represent the same line. Thus, the system is consistent and has infinitely many solutions.
### Second System
[tex]\[
\begin{array}{l}
3x - 4y = 12 \\
6x - 8y = 21
\end{array}
\][/tex]
To analyze this system, let's compare the second equation with a scaled version of the first equation. If we double the first equation:
[tex]\[
2(3x - 4y) = 2 \cdot 12 \\
6x - 8y = 24
\][/tex]
Now, compare this with the second equation from the system:
[tex]\[
6x - 8y = 21
\][/tex]
Clearly, [tex]\( 6x - 8y = 24\)[/tex] and [tex]\( 6x - 8y = 21\)[/tex] are inconsistent. There is no possible value of [tex]\( x\)[/tex] and [tex]\( y\)[/tex] that satisfies both of these equations simultaneously. Thus, the system is inconsistent or has no solutions.
### Third System
[tex]\[
\begin{array}{l}
2x - 3y = 8 \\
-3x + 2y = 8
\end{array}
\][/tex]
Let's attempt to solve this system. Multiply the first equation by 3 and the second by 2:
[tex]\[
3(2x - 3y) = 3 \cdot 8 \\
6x - 9y = 24
\][/tex]
[tex]\[
2(-3x + 2y) = 2 \cdot 8 \\
-6x + 4y = 16
\][/tex]
Now we have the system:
[tex]\[
\begin{array}{l}
6x - 9y = 24 \\
-6x + 4y = 16
\end{array}
\][/tex]
Add these two equations together:
[tex]\[
6x - 6x + (-9y + 4y) = 24 + 16 \\
-5y = 40 \\
y = -8
\][/tex]
Using [tex]\( y = -8 \)[/tex] in the first equation:
[tex]\[
2x - 3(-8) = 8 \\
2x + 24 = 8 \\
2x = 8 - 24 \\
2x = -16 \\
x = -8
\][/tex]
Thus, [tex]\( x = -8 \)[/tex] and [tex]\( y = -8 \)[/tex] is a solution, so the system is consistent and has a unique solution.
In summary:
- The first system is consistent and has a solution (infinite solutions).
- The second system is inconsistent or has infinite solutions (in this case, no solution).
- The third system is consistent and has a solution.
Conclusively filling in the blanks:
First System:
[tex]\[
\begin{array}{l}
x + 3y = 4 \\
3x + 9y = 12
\end{array}
\][/tex]
The system is consistent and has infinitely many solutions.
Second System:
[tex]\[
\begin{array}{l}
3x - 4y = 12 \\
6x - 8y = 21
\end{array}
\][/tex]
The system is inconsistent or has no solutions.
Third System:
[tex]\[
\begin{array}{l}
2x - 3y = 8 \\
-3x + 2y = 8
\end{array}
\][/tex]
The system is consistent and has a unique solution.
