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

A. [tex]\( 4x + 2 = 6 - x \)[/tex]
B. [tex]\( 5x + 2 = 6 \)[/tex]

Choose one answer:
A. Add/subtract a quantity to/from only one side
B. Add/subtract the same quantity to/from both sides
C. Multiply/divide only one side by a non-zero constant
D. Multiply/divide both sides by the same nonzero constant

2. Based on the previous answer, are the equations equivalent? In other words, do they have the same solution?

Choose one answer:
A. Yes
B. No

Sagot :

GinnyAnswer
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]
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