Let's determine the nature of the solution for the given system of equations:
[tex]\[
\begin{cases}
2h + 8k = 6 \\
-5h - 20k = -15
\end{cases}
\][/tex]
### Step 1: Simplify the equations if possible
First, observe if any simplification is required or possible for either of the equations:
1. For the first equation [tex]\( 2h + 8k = 6 \)[/tex], no simplification is needed.
2. For the second equation [tex]\( -5h - 20k = -15 \)[/tex], we can divide by [tex]\(-5\)[/tex]:
[tex]\[
h + 4k = 3
\][/tex]
### Step 2: Compare the simplified system
Now our system of equations looks like this:
[tex]\[
\begin{cases}
2h + 8k = 6 \\
h + 4k = 3
\end{cases}
\][/tex]
### Step 3: Solve the system
Let's use substitution or elimination to solve it. Here, elimination is straightforward:
Multiply the second equation [tex]\( h + 4k = 3 \)[/tex] by 2 to match the coefficients of the first equation:
[tex]\[
2(h + 4k) = 2 \times 3
\][/tex]
This results in:
[tex]\[
2h + 8k = 6
\][/tex]
Notice that our transformed second equation [tex]\(2h + 8k = 6\)[/tex] is exactly the same as the first equation.
### Step 4: Identify nature of the solution
Since both equations simplify to the same equation, they represent the same line. Two equations representing the same line indicate that every point on the line is a solution to the system. Therefore, the system does not have a single unique solution. Instead, it has infinitely many solutions.
### Conclusion
The system of equations has infinitely many solutions. Hence, the correct statement that describes the solution to the system of equations is:
The system has infinitely many solutions.
