To find the solution to the given system of equations:
[tex]\[
\begin{cases}
3y - 12x = 18 \\
2y - 8x = 12
\end{cases}
\][/tex]
we can solve these equations step by step manually.
1. Simplify each equation (if necessary):
First, let's simplify both equations by dividing them by their greatest common divisors:
For the first equation:
[tex]\[ 3y - 12x = 18, \][/tex]
divide by 3:
[tex]\[ y - 4x = 6. \][/tex]
For the second equation:
[tex]\[ 2y - 8x = 12, \][/tex]
divide by 2:
[tex]\[ y - 4x = 6.\][/tex]
Now, our simplified system is:
[tex]\[
\begin{cases}
y - 4x = 6 \\
y - 4x = 6
\end{cases}
\][/tex]
2. Analyze the simplified system:
Notice that both simplified equations are identical:
[tex]\[ y - 4x = 6. \][/tex]
This means the two original equations are actually the same equation. Therefore, there are not two separate lines that intersect at a single point. Instead, there is only one line.
As such, the system does not have a single unique solution (one pair of [tex]\((x, y)\)[/tex]). Rather, every point on the line [tex]\( y = 4x + 6 \)[/tex] is a solution to the system. Hence, there are infinitely many solutions.
3. Conclusion:
So, the correct answer to how many solutions this system has is:
B. More than 1 solution
This means the system has infinitely many solutions, as every point on the line [tex]\( y = 4x + 6 \)[/tex] is a valid solution.
