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Select the correct answer.

A system of equations and its solution are given below.

System [tex]$A$[/tex]
[tex]\[
\begin{array}{c}
5x - y = -11 \\
3x - 2y = -8 \\
\text{Solution: }(-2, 1)
\end{array}
\][/tex]

To get system [tex]$B$[/tex] below, the second equation in system [tex]$A$[/tex] was replaced by the sum of that equation and the first equation in system [tex]$A$[/tex] multiplied by -2.

System [tex]$B$[/tex]
[tex]\[
5x - y = -11
\][/tex]

A. The second equation in system [tex]$B$[/tex] is [tex]$7x = 30$[/tex]. The solution to system [tex]$B$[/tex] will not be the same as the solution to system [tex]$A$[/tex].

B. The second equation in system [tex]$B$[/tex] is [tex]$-7x = 14$[/tex]. The solution to system [tex]$B$[/tex] will not be the same as the solution to system [tex]$A$[/tex].

C. The second equation in system [tex]$B$[/tex] is [tex]$7x = 30$[/tex]. The solution to system [tex]$B$[/tex] will be the same as the solution to system [tex]$A$[/tex].

D. The second equation in system [tex]$B$[/tex] is [tex]$-7x = 14$[/tex]. The solution to system [tex]$B$[/tex] will be the same as the solution to system [tex]$A$[/tex].


Sagot :

Let us start by analyzing System A:
[tex]\[ \begin{array}{c} 5x - y = -11 \quad \text{(1)} \\ 3x - 2y = -8 \quad \text{(2)} \end{array} \][/tex]

We were instructed to obtain system B by replacing the second equation by adding it to the first equation multiplied by -2. Let's follow these steps:

1. Multiply equation (1) by -2:
[tex]\[ -2 \times (5x - y) = -2 \times (-11) \][/tex]
[tex]\[ -10x + 2y = 22 \quad \text{(3)} \][/tex]

2. Add equation (3) to equation (2):
[tex]\[ (3x - 2y) + (-10x + 2y) = -8 + 22 \][/tex]
[tex]\[ 3x - 10x = -8 + 22 \][/tex]
[tex]\[ -7x = 14 \quad \text{(4)} \][/tex]

Thus, System B is:
[tex]\[ \begin{array}{c} 5x - y = -11 \\ -7x = 14 \end{array} \][/tex]

Now, let's verify if the solution [tex]\((-2, 1)\)[/tex] is valid for both systems.

For the first equation in System A and System B:
[tex]\[ 5(-2) - 1 = -10 - 1 = -11 \][/tex]
The left-hand side equals the right-hand side, so this solution satisfies the first equation.

For the second equation in System A:
[tex]\[ 3(-2) - 2(1) = -6 - 2 = -8 \][/tex]
The left-hand side equals the right-hand side, so this solution satisfies the second equation of System A.

For the second equation in System B:
[tex]\[ -7(-2) = 14 \][/tex]
The left-hand side equals the right-hand side, so this solution satisfies the second equation of System B as well.

Hence, the solution [tex]\((-2, 1)\)[/tex] is valid for both Systems A and B. Therefore, the correct answer is:

D. The second equation in system B is [tex]\(-7x = 14\)[/tex]. The solution to system B will be the same as the solution to system A.
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