Question

Which of the following equations has exactly one solutions?
A. 5x+4=5×+4
B. -5×+4=5x+4
C. -5×+3=-5×+3
D. 5x+4=5×+3
C. 10x+3=-10×+3
D. 10x+4=10×-4

Answer 1

To determine which of the equations has exactly one solution, we can analyze each option:

A. \( 5x + 4 = 5x + 4 \)

This equation simplifies to \( 0 = 0 \) for all values of \( x \), so it has infinitely many solutions.

B. \( -5x + 4 = 5x + 4 \)

Rearranging gives:
\[
-5x - 5x = 4 - 4 \\
-10x = 0 \\
x = 0
\]
This equation has exactly one solution: \( x = 0 \).

C. \( -5x + 3 = -5x + 3 \)

This equation also simplifies to \( 0 = 0 \), so it has infinitely many solutions.

D. \( 5x + 4 = 5x + 3 \)

Rearranging gives:
\[
5x - 5x = 3 - 4 \\
0 = -1
\]
This statement is false, meaning there are no solutions.

E. \( 10x + 3 = -10x + 3 \)

Rearranging gives:
\[
10x + 10x = 3 - 3 \\
20x = 0 \\
x = 0
\]
This equation has exactly one solution: \( x = 0 \).

F. \( 10x + 4 = 10x - 4 \)

Rearranging gives:
\[
10x - 10x = -4 - 4 \\
0 = -8
\]
This statement is false, meaning there are no solutions.

In summary, the equations with exactly one solution are:

- B: \( -5x + 4 = 5x + 4 \) (solution: \( x = 0 \))
- E: \( 10x + 3 = -10x + 3 \) (solution: \( x = 0 \))

So the answer is:
**B and E (10x + 3 = -10x + 3)**.