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 -3. 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 -5. the solution to system b will 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 not be the same as the solution to system a.

Explanation:

Step1: Recall the operation for equivalent systems

When we perform a linear combination of equations (multiplying an equation by a constant and adding to another), the solution of the system remains the same. Let's analyze each option by applying the operation described.

Step2: Analyze Option A

System A:

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

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

Now, add this to the second equation of System A: \( (3x - 7y) + (-3x - 18y) = -35 + (-15) \)

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

Right side: \( -35 - 15 = -50 \)

So the new second equation is \( -25y = -50 \), which matches System B's second equation. Also, since we did a valid linear combination, the solution remains the same.

Step3: Analyze other options (briefly)

• Option B: Multiplying first equation by 7 and adding to second would not give \( -25y = -50 \). Let's check: \( 7(x + 6y)=7\times5 \Rightarrow 7x + 42y = 35 \). Adding to \( 3x - 7y = -35 \): \( 10x + 35y = 0 \), not matching. Also, it claims solution changes, which is wrong.
• Option C: Multiplying first equation by -5: \( -5x - 30y = -25 \). Adding to second equation: \( (3x - 7y) + (-5x - 30y) = -35 + (-25) \Rightarrow -2x - 37y = -60 \), not matching.
• Option D: Multiplying first equation by 3: \( 3x + 18y = 15 \). Adding to second equation: \( (3x - 7y) + (3x + 18y) = -35 + 15 \Rightarrow 6x + 11y = -20 \), not matching. Also, claims solution changes, wrong.

Answer:

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 -3. The solution to system B will be the same as the solution to system A.