Question

select the correct answer from each drop-down menu.

a system of equations and its solution are given below.

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$$\begin{aligned} &\\text{system a} \\\\ &x - y = 7 \\\\ &-3x + 9y = -39 \\\\ &\\text{solution: } (4, -3) \\end{aligned}$$

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complete the sentences to explain what steps were followed to obtain the system of equations below.

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$$\begin{aligned} &\\text{system b} \\\\ &x - y = 7 \\\\ &6y = -18 \\end{aligned}$$

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to get system b, the choose an answer equation in system a was replaced by the sum of that equation and choose an answer multiplied by the choose an answer equation. the solution to system b choose an answer the same as the solution to system a.

Explanation:

⚡ Using what you learned: Solving Systems by Elimination (Addition)

Step 1: Compare the systems

Compare System A and System B:

System A:

$$
LATEXBLOCK0
$$

System B:

$$
LATEXBLOCK1
$$

The first equation remains unchanged. The second equation in System A, \(-3x + 9y = -39\), is replaced by \(6y = -18\) in System B.

Step 2: Analyze the elimination step

To eliminate the \(x\)-term from the second equation, multiply the first equation by \(3\):

$$ 3(x - y) = 3(7) \implies 3x - 3y = 21 $$

Add this result to the second equation of System A:

$$
LATEXBLOCK2
$$

This matches the second equation in System B.

Thus, the second equation in System A was replaced by the sum of that equation and \(3\) times the first equation.

Step 3: Determine the solution equivalence

Since performing valid row operations (like adding a multiple of one equation to another) produces an equivalent system, the solution remains unchanged.

The solution to System B is the same as the solution to System A.

Answer:

To get system B, the second equation in system A was replaced by the sum of that equation and 3 times the first equation. The solution to system B is the same as the solution to system A.