Answered

Select the correct answer.

A system of equations and its solution are given below.

System A:
[tex]\[
\begin{array}{l}
-x - 2y = 7 \\
5x - 6y = -3 \\
\end{array}
\][/tex]
Solution: [tex]\((-3, -2)\)[/tex]

Choose the correct option that explains what steps were followed to obtain the system of equations below.

System B:
[tex]\[
\begin{aligned}
-x - 2y &= 7 \\
-16y &= 32
\end{aligned}
\][/tex]

A. To get System B, the second equation in System A was replaced by the sum of that equation and the first equation multiplied by 3. The solution to System B will be the same as the solution to System A.

B. To get System B, the second equation in System A was replaced by the sum of that equation and the first equation multiplied by -5. The solution to System B will not be the same as the solution to System A.

C. To get System B, the second equation in System A was replaced by the sum of that equation and the first equation multiplied by -6. The solution to System B will not be the same as the solution to System A.

D. To get System B, the second equation in System A was replaced by the sum of that equation and the first equation multiplied by 5. The solution to System B will be the same as the solution to System A.



Answer :

GinnyAnswer
Certainly! Let's go through the steps to identify which option explains how System A was transformed into System B.

First, let's look at System A:
[tex]\[ \begin{array}{l} -x - 2y = 7 \\ 5x - 6y = -3 \end{array} \][/tex]

And now, let’s look at the resulting System B:
[tex]\[ \begin{array}{l} -x - 2y = 7 \\ -16y = 32 \end{array} \][/tex]

To figure out how we got from System A to System B, we need to manipulate the second equation [tex]\(5x - 6y = -3\)[/tex] in System A.

1. Multiply the first equation by 5:
[tex]\[ 5 \cdot (-x - 2y) = 5 \cdot 7 \implies -5x - 10y = 35 \][/tex]

2. Add this result to the second equation [tex]\(5x - 6y = -3\)[/tex]:
[tex]\[ (5x - 6y) + (-5x - 10y) = -3 + 35 \][/tex]

This simplifies to:
[tex]\[ 5x - 6y - 5x - 10y = 32 \implies -16y = 32 \][/tex]

This shows that the second equation in System B results from replacing the second equation in System A with the sum of that equation and the first equation multiplied by 5.

Therefore, the correct option is:

D. To get system B, the second equation in system A was replaced by the sum of that equation and the first equation multiplied by 5. The solution to system B will be the same as the solution to system A.

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