Question

a system of equations and its solution are given below
system a
x + 6y = 5
3x - 7y = -35
solution: (-7, 2)
choose the correct option that explains what steps were followed to obtain the system of equations below.
system b
x + 6y = 5
-25y = -50
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 -5.
the solution to system b will 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 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 not be the same as the solution to system a.
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 be the same as the solution to system a.

Explanation:

Step1: Recall the operation on linear equations

When we perform operations like adding a multiple of one equation to another in a system of linear equations, the solution of the system remains unchanged (this is based on the property of equivalent systems of linear equations). Let's analyze each option by performing the suggested operation.

Step2: Analyze Option A

System A:

1. \( x + 6y = 5 \)
2. \( 3x - 7y = -35 \)

First equation multiplied by -5: \( -5(x + 6y) = -5\times5 \) which simplifies to \( -5x - 30y = -25 \)

Now, add this to the second equation of System A:
\( (3x - 7y) + (-5x - 30y) = -35 + (-25) \)
\( 3x - 7y -5x - 30y = -60 \)
\( -2x - 37y = -60 \) → This is not the second equation of System B. Wait, maybe I made a mistake. Wait, System B's second equation is \( -25y = -50 \). Wait, let's re - calculate.

Wait, first equation: \( x + 6y = 5 \), second equation: \( 3x - 7y=-35 \)

If we multiply the first equation by - 3: \( - 3x-18y=-15 \)

Add to the second equation: \( (3x - 7y)+(-3x - 18y)=-35+(-15) \)
\( 3x - 7y-3x - 18y=-50 \)
\( -25y=-50 \) → Oh, wait, maybe I misread option A. Wait, option A says "first equation multiplied by - 5". Wait, no, let's check option D.

Step3: Analyze Option D

First equation multiplied by - 3: \( -3(x + 6y)=-3\times5\) → \( -3x-18y = - 15 \)

Add this to the second equation of System A (\( 3x - 7y=-35 \)):

\( (3x - 7y)+(-3x - 18y)=-35+(-15) \)

Simplify left - hand side: \( 3x-7y - 3x-18y=-25y \)

Simplify right - hand side: \( -35 - 15=-50 \)

So we get \( -25y=-50 \), which is the second equation of System B. Also, since we are adding a multiple of one equation to another, the solution of the system remains the same.

Now let's check other options:

Option B: Multiply first equation by 7: \( 7x + 42y = 35 \)

Add to second equation: \( (3x - 7y)+(7x + 42y)=-35 + 35 \)

\( 10x + 35y = 0 \) ≠ \( -25y=-50 \)

Option C: Multiply first equation by 3: \( 3x+18y = 15 \)

Add to second equation: \( (3x - 7y)+(3x + 18y)=-35 + 15 \)

\( 6x + 11y=-20 \) ≠ \( -25y=-50 \)

Option A: Multiply first equation by - 5: \( -5x-30y=-25 \)

Add to second equation: \( (3x - 7y)+(-5x - 30y)=-35-25 \)

\( -2x-37y=-60 \) ≠ \( -25y=-50 \)

Option D: Multiply first equation by - 3: \( -3x-18y=-15 \)

Add to second equation: \( (3x - 7y)+(-3x - 18y)=-35-15 \)

\( -25y=-50 \), which is the second equation of System B. And since we performed a valid row operation (adding a multiple of one equation to another), the solution remains the same.

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 be the same as the solution to system A.