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

Select the correct statement about step 3.

A. When the equation $-3x - 5y = -5$ is subtracted from $-10x + 5y = -60$, a third linear equation, $-13x = -65$, is formed, and it has a different solution from the original equations.
B. 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.
C. When the equations $-10x + 5y = -60$ and $-3x - 5y = -5$ are added together, a third linear equation, $-13x = -65$, is formed, and it shares a common solution with the original equations.
D. When the equations $-10x + 5y = -60$ and $-3x - 5y = -5$ are added together, a third linear equation, $-13x = -65$, is formed, and it has a different solution from the original equations.

The equations − 10 x + 5 y = − 60 and − 3 x − 5 y = − 5 are added together.
This results in the equation − 13 x = − 65 .
Substituting x = 5 into − 13 x = − 65 confirms that it is a solution.
Therefore, the equation − 13 x = − 65 shares a common solution with the original equations, and the answer is C.

Explanation

Analyze the Problem We are given a system of equations and the steps to solve it. The question asks us to identify the correct statement about how Step 3 is derived from Step 2. The system of equations is:

Step 1: − 5 ( 2 x − y ) = − 5 ( 12 ) − 3 x − 5 y = − 5
Step 2: − 10 x + 5 y = − 60 − 3 x − 5 y = − 5
Step 3: − 13 x = − 65
Step 4: x = 5
Step 5: 2 ( 5 ) − y = 12
Step 6: y = − 2
Solution: ( 5 , − 2 )

Adding the Equations To get from Step 2 to Step 3, we need to combine the two equations in Step 2. Let's see what happens if we add the two equations:

( − 10 x + 5 y ) + ( − 3 x − 5 y ) = − 60 + ( − 5 ) − 10 x + 5 y − 3 x − 5 y = − 65 − 13 x = − 65
This is exactly the equation in Step 3. So, the equations in Step 2 were added together to get the equation in Step 3.

Checking the Solution Now, we need to determine if the equation in Step 3 shares a common solution with the original equations or has a different solution. The solution to the original system of equations is ( 5 , − 2 ) . Let's substitute x = 5 into the equation in Step 3:

− 13 x = − 65 − 13 ( 5 ) = − 65 − 65 = − 65
Since the equation is true when x = 5 , the equation in Step 3 shares a common solution with the original equations.

Conclusion Therefore, the correct statement is: When the equations − 10 x + 5 y = − 60 and − 3 x − 5 y = − 5 are added together, a third linear equation, − 13 x = − 65 , is formed, and it shares a common solution with the original equations.

Examples
Systems of equations are used in many real-world applications, such as determining the break-even point for a business. For example, if a company has fixed costs of $10,000 and variable costs of $5 per unit, and sells each unit for 15 , w ec an se t u p a sys t e m o f e q u a t i o n s t o f in d t h e n u mb ero f u ni t s t ha t n ee d t o b eso l d t o b re ak e v e n . L e t x b e t h e n u mb ero f u ni t sso l d an d y b e t h e t o t a l cos t / re v e n u e . T h ecos t e q u a t i o ni s y = 5x + 10000 an d t h ere v e n u ee q u a t i o ni s y = 15x$. Solving this system of equations will give the break-even point. Understanding how to solve systems of equations is crucial for making informed business decisions.

Answered by GinnyAnswer | 2025-07-03

In Step 3, adding the equations − 10 x + 5 y = − 60 and − 3 x − 5 y = − 5 results in − 13 x = − 65 . This new equation shares a common solution with the original equations, specifically x = 5 and y = − 2 . Therefore, the correct answer is option C.
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Answered by Anonymous | 2025-07-04