Answer :
Sure! Let's work through the given system of equations step by step to find the equivalent system.
The original system is:
[tex]\[
\begin{aligned}
6r + 4s & = 8 \\
2(r + 5s) & = 2s
\end{aligned}
\][/tex]
First, we'll simplify the second equation. Expand and simplify it:
[tex]\[
2(r + 5s) = 2s \\
2r + 10s = 2s
\][/tex]
Subtract [tex]\(2s\)[/tex] from both sides:
[tex]\[
2r + 8s = 0
\][/tex]
Now we have the simplified system:
[tex]\[
\begin{aligned}
6r + 4s & = 8 \\
2r + 8s & = 0
\end{aligned}
\][/tex]
Next, we need to rewrite [tex]\(6r + 4s = 8\)[/tex] and [tex]\(2r + 8s = 0\)[/tex] in an equivalent form. We know from the problem description that we are looking for systems in a specific answer format.
Let's look at the possible choices and check which system matches our simplified system:
1. [tex]\[
\begin{aligned}
8r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
2. [tex]\[
\begin{aligned}
2r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
3. [tex]\[
\begin{aligned}
4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
4. [tex]\[
\begin{aligned}
-4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
We can see that only the third choice conforms to the structure derived from the original system while accounting for the transformations needed to simplify:
1. [tex]\( 6r + 4s = 8 \implies 4(r + s) = 8 \implies 4r + 4s = 8 \)[/tex]
2. [tex]\( 2r + 8s = 0 \text{ already translates directly to } 8s = 0\)[/tex]
Thus, the equivalent system is:
[tex]\[
\begin{aligned}
4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
Therefore, the correct answer is indeed:
[tex]\[
\begin{aligned}
4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
The original system is:
[tex]\[
\begin{aligned}
6r + 4s & = 8 \\
2(r + 5s) & = 2s
\end{aligned}
\][/tex]
First, we'll simplify the second equation. Expand and simplify it:
[tex]\[
2(r + 5s) = 2s \\
2r + 10s = 2s
\][/tex]
Subtract [tex]\(2s\)[/tex] from both sides:
[tex]\[
2r + 8s = 0
\][/tex]
Now we have the simplified system:
[tex]\[
\begin{aligned}
6r + 4s & = 8 \\
2r + 8s & = 0
\end{aligned}
\][/tex]
Next, we need to rewrite [tex]\(6r + 4s = 8\)[/tex] and [tex]\(2r + 8s = 0\)[/tex] in an equivalent form. We know from the problem description that we are looking for systems in a specific answer format.
Let's look at the possible choices and check which system matches our simplified system:
1. [tex]\[
\begin{aligned}
8r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
2. [tex]\[
\begin{aligned}
2r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
3. [tex]\[
\begin{aligned}
4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
4. [tex]\[
\begin{aligned}
-4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
We can see that only the third choice conforms to the structure derived from the original system while accounting for the transformations needed to simplify:
1. [tex]\( 6r + 4s = 8 \implies 4(r + s) = 8 \implies 4r + 4s = 8 \)[/tex]
2. [tex]\( 2r + 8s = 0 \text{ already translates directly to } 8s = 0\)[/tex]
Thus, the equivalent system is:
[tex]\[
\begin{aligned}
4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
Therefore, the correct answer is indeed:
[tex]\[
\begin{aligned}
4r + 4s & = 8 \\
8s & = 0
\end{aligned}
\][/tex]
