Let’s solve the problem provided in the image.

Question:

Which equation has infinitely many solutions?

To have infinitely many solutions, an equation must simplify to an identity, such as $0 = 0$. This happens when both sides of the equation are equivalent after simplification.

Analyze the options:

(A) $12 + 4x = 6x + 10 - 2x$

1. Combine like terms on the right: $6x - 2x + 10 = 4x + 10$
2. The equation becomes: $12 + 4x = 4x + 10$
3. Subtract $4x$ from both sides: $12 = 10$ This is a contradiction, so no solution.

(B) $5x + 14 - 4x = 23 + x - 9$

1. Simplify both sides:
   • Left: $5x - 4x + 14 = x + 14$
   • Right: $23 - 9 + x = x + 14$
2. The equation becomes: $x + 14 = x + 14$ This is true for all $x$, so this equation has infinitely many solutions.

(C) $x + 9 - 0.8x = 5.2x + 17 - 8$

1. Simplify both sides:
   • Left: $x - 0.8x + 9 = 0.2x + 9$
   • Right: $5.2x + 17 - 8 = 5.2x + 9$
2. The equation becomes: $0.2x + 9 = 5.2x + 9$
3. Subtract $9$ from both sides: $0.2x = 5.2x$
4. Subtract $0.2x$: $0 = 5x$ This is false unless $x = 0$, so not infinitely many solutions.

(D) $4x - 2x = 20$

1. Simplify the left side: $2x = 20$
2. Solve for $x$: $x = 10$ This is a single solution, so not infinitely many solutions.

Answer:

(B) $5x + 14 - 4x = 23 + x - 9$

Would you like a detailed explanation of any step or have additional questions?

Related Questions:

1. How can you determine when an equation has no solution?
2. What is the difference between equations with infinitely many solutions and a single solution?
3. How do you simplify equations with variables on both sides?
4. Why does $12 + 4x = 6x + 10 - 2x$ result in a contradiction?
5. Can you create your own equation that has infinitely many solutions?

Tip:

Always simplify both sides of an equation as much as possible to identify contradictions or identities!