Let's break down the problem step-by-step:
### Part 1: How to Derive Equation B from Equation A
Given:
- Equation A: [tex]\( 3(x + 2) = 18 \)[/tex]
- Equation B: [tex]\( x + 2 = 18 \)[/tex]
We need to determine how Equation B can be derived from Equation A. Let's analyze the steps required:
1. Simplify Equation A using the distributive property:
[tex]\( 3(x + 2) = 18 \)[/tex]
Applying the distributive property:
[tex]\( 3 \cdot x + 3 \cdot 2 = 18 \)[/tex]
Which simplifies to:
[tex]\( 3x + 6 = 18 \)[/tex]
2. Isolate the variable [tex]\(x\)[/tex]:
To solve for [tex]\( x \)[/tex], we need to isolate it on one side of the equation. We'll subtract 6 from both sides:
[tex]\( 3x + 6 - 6 = 18 - 6 \)[/tex]
This simplifies to:
[tex]\( 3x = 12 \)[/tex]
Next, we divide by 3:
[tex]\( \frac{3x}{3} = \frac{12}{3} \)[/tex]
Which simplifies to:
[tex]\( x = 4 \)[/tex]
However, this process does not match Equation B. Instead, it appears that:
- Equation B is not derived from Equation A by any direct, simple algebraic manipulation.
Given the options:
- (A) Multiply/divide both sides by the same non-zero constant
- (B) Multiply/divide only one side by a non-zero constant
- (C) Rewrite one side (or both) by combining like terms
- (D) Rewrite one side (or both) using the distributive property
The correct choice is:
### Answer 1: (A) Multiply/divide both sides by the same non-zero constant
### Part 2: Are the Equations Equivalent?
To determine if the equations are equivalent, we need to check if they have the same solution set.
Solve Equation B:
Equation B: [tex]\( x + 2 = 18 \)[/tex]
Subtract 2 from both sides:
[tex]\( x = 18 - 2 \)[/tex]
Which simplifies to:
[tex]\( x = 16 \)[/tex]
Given the solutions:
- Equation A solution: [tex]\( x = 4 \)[/tex]
- Equation B solution: [tex]\( x = 16 \)[/tex]
The solutions are different, meaning the equations are not equivalent.
Given the options:
- (A) Yes
- (B) No
The correct choice is:
### Answer 2: (B) No
Thus, the complete process shows the steps leading to the answers:
1) (A) Multiply/divide both sides by the same non-zero constant
2) (B) No
