Certainly! Let's understand the steps involved in transforming System A to System B.
### System A
Given:
[tex]\[
\begin{aligned}
x - y &= 3 \\
-2x + 4y &= -2
\end{aligned}
\][/tex]
### System B
Given:
[tex]\[
\begin{aligned}
x - y &= 3 \\
2x &= 10
\end{aligned}
\][/tex]
Explanation:
1. To transform System A into System B, the first equation remains the same:
[tex]\[
x - y = 3
\][/tex]
2. For the second equation of System B, we derive it from the second equation of System A. The key step here involves multiplying the second equation of System A by 0.5:
[tex]\[
-2x + 4y = -2 \quad \text{becomes} \quad -x + 2y = -1
\][/tex]
3. Add this modified equation to the first equation of System A:
[tex]\[
(x - y) + (-x + 2y) = 3 + (-1)
\][/tex]
Simplifying this gives:
[tex]\[
y = 2
\][/tex]
4. Substitute [tex]\(y = 2\)[/tex] back into the first equation of System A to solve for [tex]\(x\)[/tex]:
[tex]\[
x - 2 = 3 \quad \Rightarrow \quad x = 5
\][/tex]
5. Verify by substituting [tex]\(x = 5\)[/tex] and [tex]\(y = 2\)[/tex] into the system, we get:
[tex]\[
2x = 2(5) = 10
\][/tex]
Thus, the second equation of System B, [tex]\(2x = 10\)[/tex], holds true.
### Completing the Sentences:
To get system B, the first [tex]\( \boxed{\text{equation}} \)[/tex] in system A was replaced by the sum of that equation and the second equation multiplied by [tex]\( \boxed{0.5} \)[/tex]. The solution to system B [tex]\( \boxed{\text{is}} \)[/tex] the same as the solution to system A.
### Final Answer:
1. To get system B, the first equation in system A was replaced by the sum of that equation and the second equation multiplied by 0.5.
2. The solution to system B is the same as the solution to system A.
