kylorensimp
Answered

Select the correct answer.

Consider the following system of equations:
[tex]
\begin{aligned}
2x - y &= 12 \\
-3x - 5y &= -5
\end{aligned}
[/tex]

The steps for solving the given system of equations are shown below:
[tex]
\begin{array}{l}
\text{Step 1:} \quad -5(2x - y) = -5(12) \\
-3x - 5y = -5 \\
\text{Step 2:} \quad -10x + 5y = -60 \\
-3x - 5y = -5 \\
\text{Step 3:} \quad -13x = -65 \\
\text{Step 4:} \quad x = 5 \\
\text{Step 5:} \quad 2(5) - y = 12 \\
\text{Step 6:} \quad y = -2 \\
\text{Solution:} \quad (5, -2) \\
\end{array}
[/tex]

Select the correct statement about Step 3:

A. When the equation [tex]-3x - 5y = -5[/tex] is subtracted from [tex]-10x + 5y = -60[/tex], a third linear equation, [tex]-13x = -65[/tex], is formed, and it shares a common solution with the original equations.

B. When the equation [tex]-3x - 5y = -5[/tex] is subtracted from [tex]-10x + 5y = -60[/tex], a third linear equation, [tex]-13x = -65[/tex], is formed, and it has a different solution from the original equations.

C. When the equations [tex]-10x + 5y = -60[/tex] and [tex]-3x - 5y = -5[/tex] are added together, a third linear equation, [tex]-13x = -65[/tex], is formed, and it has a different solution from the original equations.

D. When the equations [tex]-10x + 5y = -60[/tex] and [tex]-3x - 5y = -5[/tex] are added together, a third linear equation, [tex]-13x = -65[/tex], is formed, and it shares a common solution with the original equations.



Answer :

GinnyAnswer
Given the system of equations:
[tex]\[ \begin{aligned} 2x - y &= 12 \\ -3x - 5y &= -5 \end{aligned} \][/tex]

Follow the steps to solve the system:

1. Step 1: Multiply the first equation by -5:
[tex]\[ -5(2x - y) = -5(12) \][/tex]
which results in:
[tex]\[ -10x + 5y = -60 \][/tex]

Now our system is:
[tex]\[ \begin{aligned} -10x + 5y &= -60 \\ -3x - 5y &= -5 \end{aligned} \][/tex]

2. Step 2: Rewrite the system with the new equation from Step 1:
[tex]\[ \begin{aligned} -10x + 5y &= -60 \\ -3x - 5y &= -5 \end{aligned} \][/tex]

3. Step 3: Subtract the second equation from the first:
[tex]\[ (-10x + 5y) - (-3x - 5y) = -60 - (-5) \][/tex]
Simplifying this, you get:
[tex]\[ -10x + 5y + 3x + 5y = -60 + 5 \][/tex]
[tex]\[ -7x + 10y = -55 \][/tex]
Since the first variable terms combine to [tex]\(-7x\)[/tex] and the constants on the right-hand side combine to [tex]\(-55\)[/tex], you end up with:
[tex]\[ -7x = -55 \][/tex]
This simplifies to:
[tex]\[ x = 5 \][/tex]

4. Now, substitute [tex]\(x = 5\)[/tex] back into one of the original equations to find [tex]\(y\)[/tex]. Choose [tex]\(2x - y = 12\)[/tex]:
[tex]\[ 2(5) - y = 12 \][/tex]
[tex]\[ 10 - y = 12 \][/tex]
[tex]\[ -y = 2 \][/tex]
[tex]\[ y = -2 \][/tex]

Thus, the solution to the original system of equations is [tex]\((x, y) = (5, -2)\)[/tex].

Given the steps and the solution, the correct statement about Step 3 is:
A. When the equation [tex]\( -3x - 5y = -5 \)[/tex] is subtracted from [tex]\( -10x + 5y = -60 \)[/tex], a third linear equation, [tex]\( -13x = -65 \)[/tex], is formed, and it shares a common solution with the original equations.

Therefore, the correct answer is:
A

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