Let's analyze the problem provided in the image.

System A: $-x - 2y = 7$ $5x - 6y = -3$ Solution: $(-3, -2)$

System B: $-x - 2y = 7$ $-16y = 32$

We need to determine which of the options correctly explains the transformation steps that yield System B from System A.

Step-by-Step Analysis:

1. The first equation in System B, $-x - 2y = 7$, is identical to the first equation in System A, so there’s no modification to it.

2. The second equation in System B, $-16y = 32$, likely results from combining or transforming the equations in System A.

3. In System A, the second equation is $5x - 6y = -3$. To obtain $-16y = 32$, we could attempt to eliminate $x$ by using the first equation.

4. Multiply the first equation by 5 to match the coefficient of $x$ in the second equation of System A: $5(-x - 2y) = 5 \times 7 \Rightarrow -5x - 10y = 35$

5. Now, add this equation to the second equation of System A: $(5x - 6y) + (-5x - 10y) = -3 + 35$ $-16y = 32$

This transformation confirms that System B is derived by adding the second equation in System A to the first equation in System A multiplied by 5, which results in $-16y = 32$.

Conclusion:

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

Would you like further clarification on any part of this solution, or perhaps explore more about systems of linear equations?

Additional Questions for Exploration:

1. What are the different methods for solving systems of linear equations?
2. How can you verify if two systems of equations have the same solution?
3. What does it mean for two systems of equations to be equivalent?
4. How does matrix representation help in solving systems of equations?
5. Can we apply similar transformations to non-linear systems of equations?

Tip:

When working with systems of equations, always check if you can simplify or eliminate variables by combining equations, as this can often reveal the solution quickly.