Let's carefully analyze each question as you requested.
### Question 1: How can we get Equation [tex]\( B \)[/tex] from Equation [tex]\( A \)[/tex]?
Given Equations:
- [tex]\( A: 3(x + 2) = 18 \)[/tex]
- [tex]\( B: x + 2 = 6 \)[/tex]
To transform Equation [tex]\( A \)[/tex] into Equation [tex]\( B \)[/tex], we need to isolate the same expression on both sides of the equation.
Starting with Equation [tex]\( A: 3(x + 2) = 18 \)[/tex]:
1. Divide both sides of the equation by 3:
[tex]\[
\frac{3(x + 2)}{3} = \frac{18}{3}
\][/tex]
2. Simplifying both sides, we get:
[tex]\[
x + 2 = 6
\][/tex]
Thus, we can see that Equation [tex]\( B \)[/tex] is obtained by dividing both sides of Equation [tex]\( A \)[/tex] by the non-zero constant 3.
Therefore, the correct choice is:
- C: Multiply/divide both sides by the same non-zero constant.
### Question 2: Are the equations equivalent? In other words, do they have the same solution?
To determine if the equations are equivalent, we need to find the solutions for both equations and see if they match.
Solving Equation [tex]\( B \)[/tex]:
[tex]\[
x + 2 = 6
\][/tex]
Subtract 2 from both sides:
[tex]\[
x = 6 - 2
\][/tex]
[tex]\[
x = 4
\][/tex]
Solving Equation [tex]\( A \)[/tex]:
[tex]\[
3(x + 2) = 18
\][/tex]
Divide both sides by 3:
[tex]\[
x + 2 = 6
\][/tex]
Now, we have the same equation as [tex]\( B \)[/tex], and we know from above that:
[tex]\[
x = 4
\][/tex]
Since both equations simplify to the same solution [tex]\( x = 4 \)[/tex], the equations are equivalent.
Therefore, the correct choice is:
- A: Yes, the equations are equivalent and have the same solution.
So the answers to the questions are:
1. C: Multiply/divide both sides by the same non-zero constant.
2. A: Yes, the equations are equivalent and have the same solution.
