Let's solve the given problem step-by-step:
### 1. Transforming Equation [tex]\(A\)[/tex] into Equation [tex]\(B\)[/tex]
Consider the given equations:
[tex]\[ A: \quad 4x + 2 = 6 - x \][/tex]
[tex]\[ B: \quad 5x + 2 = 6 \][/tex]
We need to determine how to transform Equation [tex]\(A\)[/tex] into Equation [tex]\(B\)[/tex].
Starting from Equation [tex]\(A\)[/tex]:
[tex]\[ 4x + 2 = 6 - x \][/tex]
Add [tex]\(x\)[/tex] to both sides to combine the [tex]\(x\)[/tex] terms:
[tex]\[ 4x + x + 2 = 6 - x + x \][/tex]
This simplifies to:
[tex]\[ 5x + 2 = 6 \][/tex]
Therefore, to transform Equation [tex]\(A\)[/tex] into Equation [tex]\(B\)[/tex], we need to add the same quantity (which is [tex]\(x\)[/tex]) to both sides.
So, the correct answer to how we transform Equation [tex]\(A\)[/tex] into Equation [tex]\(B\)[/tex] is:
[tex]\[ \text{(B) Add/subtract the same quantity to/from both sides} \][/tex]
### 2. Checking if Equations are Equivalent
Next, we need to determine whether the two equations are equivalent, meaning they have the same solution.
We solve both equations:
#### Solving Equation [tex]\(A\)[/tex]:
[tex]\[ 4x + 2 = 6 - x \][/tex]
Add [tex]\(x\)[/tex] to both sides:
[tex]\[ 4x + x + 2 = 6 \][/tex]
[tex]\[ 5x + 2 = 6 \][/tex]
Subtract 2 from both sides:
[tex]\[ 5x = 4 \][/tex]
Divide by 5:
[tex]\[ x = \frac{4}{5} \][/tex]
#### Solving Equation [tex]\(B\)[/tex]:
[tex]\[ 5x + 2 = 6 \][/tex]
Subtract 2 from both sides:
[tex]\[ 5x = 4 \][/tex]
Divide by 5:
[tex]\[ x = \frac{4}{5} \][/tex]
Since both equations lead to the same solution ([tex]\( x = \frac{4}{5} \)[/tex]), the equations are equivalent.
So, the correct answer to whether the equations are equivalent is:
[tex]\[ \text{(A) Yes} \][/tex]
### Final Answers:
1) How can we get Equation [tex]\(B\)[/tex] from Equation [tex]\(A\)[/tex]?
[tex]\[ \text{(B) Add/subtract the same quantity to/from both sides} \][/tex]
2) Based on the previous answer, are the equations equivalent?
[tex]\[ \text{(A) Yes} \][/tex]
