faithannr123
Answered

System A
[tex]\[
\begin{array}{l}
-x-2y = 7 \\
5x-6y = -3 \\
\text{Solution: } (-3, -2)
\end{array}
\][/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 5. 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 3. The solution to System B will be the same as the solution to System A.



Answer :

GinnyAnswer
Let's analyze the transformations to go from System A to System B step-by-step.

System A:
[tex]\[ \begin{array}{l} -x - 2y = 7 \quad \text{(Equation 1)}\\ 5x - 6y = -3 \quad \text{(Equation 2)} \end{array} \][/tex]

System B:
[tex]\[ \begin{array}{l} -x - 2y = 7 \quad \text{(Equation 1)}\\ -16y = 32 \quad \text{(Equation 2')} \end{array} \][/tex]

We are given four options to identify how Equation 2 in System A was transformed to Equation 2' in System B:

### Testing Option A:
> A. The second equation in system A was replaced by the sum of that equation and the first equation multiplied by 5.

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

2. Add this result to the second equation in System A:
[tex]\[ (5x - 6y) + (-5x - 10y) = -3 + 35 \][/tex]
[tex]\[ -16y = 32 \][/tex]

This matches Equation 2' in System B. Thus, option A describes the correct transformation.

### Verifying other options:

#### Option B:
> B. The second equation in system A was replaced by the sum of that equation and the first equation multiplied by -5.

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

2. Add this result to the second equation in System A:
[tex]\[ (5x - 6y) + (5x + 10y) = -3 - 35 \][/tex]
[tex]\[ 10x + 4y = -38 \][/tex]

This does not match Equation 2' in System B.

#### Option C:
> C. The second equation in system A was replaced by the sum of that equation and the first equation multiplied by -6.

1. Multiply the first equation by -6:
[tex]\[ -6(-x - 2y) = -6 \times 7 \implies 6x + 12y = -42 \][/tex]

2. Add this result to the second equation in System A:
[tex]\[ (5x - 6y) + (6x + 12y) = -3 - 42 \][/tex]
[tex]\[ 11x + 6y = -45 \][/tex]

This does not match Equation 2' in System B.

#### Option D:
> D. The second equation in system A was replaced by the sum of that equation and the first equation multiplied by 3.

1. Multiply the first equation by 3:
[tex]\[ 3(-x - 2y) = 3 \times 7 \implies -3x - 6y = 21 \][/tex]

2. Add this result to the second equation in System A:
[tex]\[ (5x - 6y) + (-3x - 6y) = -3 + 21 \][/tex]
[tex]\[ 2x - 12y = 18 \][/tex]

This does not match Equation 2' in System B.

### Conclusion:
The correct option is:
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 5. The solution to system B will be the same as the solution to system A.

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