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Which of the statements is true for the two equations below?

Equation A: [tex]6 + 3x = 3x - 3[/tex]

Equation B: [tex]2(4x - 1) = 8x - 2[/tex]

A. Equation A and Equation B have an infinite number of solutions.
B. Equation A has no solution and Equation B has an infinite number of solutions.
C. Equation A has an infinite number of solutions and Equation B has no solution.
D. Equation A and Equation B have no solution.

Sagot :

GinnyAnswer
Let's analyze each of the equations step-by-step to determine the nature of their solutions.

### Equation A: \( 6 + 3x = 3x - 3 \)
1. Initial equation:

[tex]\[ 6 + 3x = 3x - 3 \][/tex]

2. Subtract \(3x\) from both sides:

[tex]\[ 6 + 3x - 3x = 3x - 3 - 3x \][/tex]

3. Simplify:

[tex]\[ 6 = -3 \][/tex]

The statement \(6 = -3\) is a contradiction, indicating there is no solution for Equation A. Therefore, Equation A has no solution.

### Equation B: \( 2(4x - 1) = 8x - 2 \)
1. Distribute the 2 on the left-hand side:

[tex]\[ 2 \cdot 4x - 2 \cdot 1 = 8x - 2 \][/tex]

2. Simplify:

[tex]\[ 8x - 2 = 8x - 2 \][/tex]

This simplifies to an identity, which is \( 8x - 2 = 8x - 2 \). Since both sides of the equation are always equal regardless of the value of \( x \), Equation B has an infinite number of solutions.

### Summary of the Solutions:
- Equation A has no solution.
- Equation B has an infinite number of solutions.

### Verify the Given Statements:
1. Equation \( A \) and Equation \( B \) have an infinite number of solutions.
- This is false because Equation A has no solution.
2. Equation \( A \) has no solution and Equation \( B \) has an infinite number of solutions.
- This is true.
3. Equation \( A \) has an infinite number of solutions and Equation \( B \) has no solution.
- This is false because it is the reverse of the actual solution.
4. Equation \( A \) and Equation \( B \) have no solution.
- This is false because Equation B has an infinite number of solutions.

Based on the above analysis, the true statement is:

Equation \( A \) has no solution and Equation \( B \) has an infinite number of solutions.

Thus, the correct answer is 2.
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