Question

Which equation has infinitely many solutions?
Responses

A. 8x+3/2= 2x + 4

B. 15x+5/3

C. 8x+5/2 = 8x + 6

D. 9x+6/3 = 3x + 2

Answer 1

To determine which equation has infinitely many solutions, we need to check if the equations are equivalent or simplify into true statements.

Let's check each option:

**A.** \( 8x + \frac{3}{2} = 2x + 4 \)

Rearranging gives:
\[
8x - 2x = 4 - \frac{3}{2}
\]
This simplifies to:
\[
6x = \frac{8}{2} - \frac{3}{2} = \frac{5}{2}
\]
Thus,
\[
x = \frac{5}{12}
\]
This has a single solution.

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**B.** \( 15x + \frac{5}{3} \)

This is not an equation since there is no equal sign, so we cannot determine if it has solutions.

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**C.** \( 8x + \frac{5}{2} = 8x + 6 \)

If we subtract \( 8x \) from both sides, we get:
\[
\frac{5}{2} = 6
\]
This statement is false (since \( \frac{5}{2} = 2.5 \)), meaning this equation has no solutions.

---

**D.** \( \frac{9x + 6}{3} = 3x + 2 \)

Simplifying the left side:
\[
3x + 2 = 3x + 2
\]
This simplifies to the identity \( 3x + 2 = 3x + 2 \), which is always true for all \( x \). Thus, this equation has infinitely many solutions.

**Conclusion:** The equation with infinitely many solutions is **D**.