Question

solving systems using elimination: mastery test

\\
\

$$\begin{aligned} 2x - y &= 12 \\\\ -3x - 5y &= -5 \\end{aligned}$$

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the steps for solving the given system of equations are shown below.

step 1:
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$$\begin{aligned} -5(2x - y) &= -5(12) \\\\ -3x - 5y &= -5 \\end{aligned}$$

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step 2:
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$$\begin{aligned} -10x + 5y &= -60 \\\\ -3x - 5y &= -5 \\end{aligned}$$

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step 3:
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-13x = -65
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step 4:
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x = 5
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step 5:
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2(5) - y = 12
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step 6:
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y = -2
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solution: \\((5, -2)\\)

select the correct statement about step 3.

when the equation \\(-3x - 5y = -5\\) is subtracted from \\(-10x + 5y = -60\\), a third linear equation, \\(-13x = -65\\), is formed, and it shares a common solution with the original equations.

Explanation:

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

Step 1: Analyze Step 2 to Step 3

In Step 2, we have the system:

$$
LATEXBLOCK0
$$

In Step 3, these two equations are added together:

$$
LATEXBLOCK1
$$

Step 2: Evaluate the Statement

Adding the two equations together eliminates the variable \(y\) and produces a new linear equation, \(-13x = -65\), which shares the same solution set as the original system.

Therefore, the correct statement describing Step 3 is:

"When the equation \(-3x - 5y = -5\) is added to \(-10x + 5y = -60\), a third linear equation, \(-13x = -65\), is formed, and it shares a common solution with the original equations."

Answer:

When the equation \(-3x - 5y = -5\) is added to \(-10x + 5y = -60\), a third linear equation, \(-13x = -65\), is formed, and it shares a common solution with the original equations.