Question

Candice wrote the four equations below. She examined them, without solving them, to determine which equation has an infinite number of solutions.

Negative 5 x + 1 = negative 5 x + 1. Negative 2 x + 1 = negative 2 x + 2. 3 x + 5 = 3 x minus 5. 4 x minus 2 = x + 1.

Which of Candice’s equations has an infinite number of solutions?
Negative 5 x + 1 = negative 5 x + 1
Negative 2 x + 1 = negative 2 x + 2
3 x + 5 = 3 x minus 5

Answer 1

An equation has an infinite number of solutions if both sides of the equation are identical, meaning that any value of x will satisfy the equation. Let's analyze Candice's equations one by one:

1. **Negative 5 x + 1 = negative 5 x + 1**
Both sides are identical, so this equation holds true for all values of x. This means it has an infinite number of solutions.

2. **Negative 2 x + 1 = negative 2 x + 2**
If we simplify this, we see that subtracting negative 2x from both sides gives us 1 = 2, which is false. This equation has no solutions.

3. **3 x + 5 = 3 x - 5**
Simplifying this by subtracting 3x from both sides gives us 5 = -5, which is also false. This means there are no solutions.

4. **4 x - 2 = x + 1**
Rearranging gives us 4x - x = 1 + 2, or 3x = 3, leading to x = 1, which is a single solution (not infinite).

Therefore, the only equation from Candice's list that has an infinite number of solutions is:

**Negative 5 x + 1 = negative 5 x + 1**