Question

system a
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$$\begin{cases} x + 6y = 5 \\\\ 3x - 7y = -35 \\end{cases}$$

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solution: \\((-7,2)\\)

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

system b
\\\

$$\begin{cases} x + 6y = 5 \\\\ -25y = -50 \\end{cases}$$

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• 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 not be the same as the solution to system 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.

Explanation:

Analyze the original system of equations

We are given the original system of equations, which we will call System A:

$$
LATEXBLOCK0
$$

The solution to this system is given as \((-7, 2)\).

Analyze the transformed system

We are given System B:

$$
LATEXBLOCK1
$$

We observe that the first equation remains unchanged: \(E1' = E1\).
The second equation has been replaced by a new equation: \(E2' = -25y = -50\).

Determine the algebraic transformation

Using the Elimination Method for Systems of Linear Equations, we look for a linear combination of \(E1\) and \(E2\) that eliminates the variable \(x\).
To eliminate \(x\), we can multiply \(E1\) by \(-3\) and add it to \(E2\):

$$ -3(E1) + E2 \implies -3(x + 6y) + (3x - 7y) = -3(5) + (-35) $$

Simplifying this expression:

$$ -3x - 18y + 3x - 7y = -15 - 35 $$

$$ -25y = -50 $$

This matches the second equation in System B exactly.
Thus, the second equation in System A was replaced by the sum of that equation (\(E2\)) and the first equation (\(E1\)) multiplied by \(-3\).

Evaluate the equivalence of the systems

Since System B was obtained from System A using a valid row operation (replacing an equation with a linear combination of itself and another equation), the two systems are equivalent.
This means they share the exact same solution set.
Therefore, the solution to System B is the same as the solution to System A.

Match with the correct option

Let's evaluate the visible options:

• Option 1 states: "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 not be the same as the solution to system A." (Incorrect, multiplier is \(-3\), and solutions are the same).
• Let's reconstruct the correct option based on standard algebra curriculum choices for this problem:

"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."
Let's represent the options clearly in the final answer.

Answer:

• (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 not 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 7. The solution to system B will not be the same as the solution to system 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 -3. The solution to system B will be the same as the solution to system A. (Correct answer)
• (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 not be the same as the solution to system A.