Answer the following questions about Equations A and B:

A. [tex]\(\frac{x}{4} + 1 = -3\)[/tex]
B. [tex]\(x + 4 = -12\)[/tex]

1) How can we get Equation B from Equation A?

A. Rewrite one side (or both) using the distributive property
B. Rewrite one side (or both) by combining like terms
C. Multiply/divide only one side by a non-zero constant
D. Multiply/divide both sides by the same non-zero constant

2) Based on the previous answer, are the equations equivalent?



Answer :

Let's answer the given questions step-by-step.

### Given Equations:
- Equation A: [tex]\(\frac{x}{4} + 1 = -3\)[/tex]
- Equation B: [tex]\(x + 4 = -12\)[/tex]

### Question 1: How can we get Equation B from Equation A?

Let's solve Equation A for [tex]\(x\)[/tex] and see if we can transform it into Equation B.

#### Step-by-Step Solution for Equation A:

1. Start with Equation A:
[tex]\[ \frac{x}{4} + 1 = -3 \][/tex]

2. Subtract 1 from both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[ \frac{x}{4} = -3 - 1 \][/tex]
[tex]\[ \frac{x}{4} = -4 \][/tex]

3. Multiply both sides by 4 to solve for [tex]\(x\)[/tex]:
[tex]\[ x = -4 \times 4 \][/tex]
[tex]\[ x = -16 \][/tex]

Now that we have [tex]\(x = -16\)[/tex], let's substitute [tex]\(x\)[/tex] into Equation B to check if we get the same equation.

4. Substitute [tex]\(x = -16\)[/tex] into Equation B:
[tex]\[ x + 4 = -12 \][/tex]
[tex]\[ -16 + 4 = -12 \][/tex]
[tex]\[ -12 = -12 \][/tex]

This shows that the value of [tex]\(x\)[/tex] from Equation A satisfies Equation B.

### Choosing the Transformation Method:

From the solving process, it is clear that:

- We multiplied both sides of Equation A by a constant (non-zero) value. This means the correct method is:

[tex]\[ \boxed{\text{D) Multiply/divide both sides by the same nonzero constant}} \][/tex]

### Question 2: Are the equations equivalent?

Based on the transformation method chosen above and the calculations, we verified that:

- The value [tex]\(x = -16\)[/tex] obtained from Equation A satisfies Equation B.
- Both equations, after proper manipulation, result in the same [tex]\(x\)[/tex] value and maintain equality when simplifying properly.

Therefore, the equations are indeed equivalent.

[tex]\[ \boxed{\text{Yes, the equations are equivalent.}} \][/tex]