Question

Question 1

Examine the six different equations that follow. Determine which equations have one solution, no solution, or infinitely many solutions.

Put a circle around the two equations that have infinite solutions. Put a square around the two equations that have no solution.

Underline the two equations that have one solution.

2x + 3 = 2x + 7

7x + 3 = 7x + 3

4x + 22 = 8x + 10

5x + 5 = 5x + 4

3x + 18 = 5x + 8

3(4x + 8) = 4(3x + 6)

Answer 1

Let's analyze each of the equations:

1. **2x + 3 = 2x + 7**
Subtracting 2x from both sides:
\(3 = 7\) (This is false)
**No solution. (Square)**

2. **7x + 3 = 7x + 3**
Both sides are equal regardless of the value of x.
**Infinitely many solutions. (Circle)**

3. **4x + 22 = 8x + 10**
Subtracting 4x from both sides:
\(22 = 4x + 10\)
\(12 = 4x\)
\(x = 3\) (This has one solution)
**One solution. (Underline)**

4. **5x + 5 = 5x + 4**
Subtracting 5x from both sides:
\(5 = 4\) (This is false)
**No solution. (Square)**

5. **3x + 18 = 5x + 8**
Rearranging gives:
\(18 - 8 = 5x - 3x\)
\(10 = 2x\)
\(x = 5\) (This has one solution)
**One solution. (Underline)**

6. **3(4x + 8) = 4(3x + 6)**
Expanding both sides gives:
\(12x + 24 = 12x + 24\)
Both sides are equal regardless of the value of x.
**Infinitely many solutions. (Circle)**

In summary:

- Equations with **infinitely many solutions**:
\(7x + 3 = 7x + 3\) (Circle)
\(3(4x + 8) = 4(3x + 6)\) (Circle)

- Equations with **no solution**:
\(2x + 3 = 2x + 7\) (Square)
\(5x + 5 = 5x + 4\) (Square)

- Equations with **one solution**:
\(4x + 22 = 8x + 10\) (Underline)
\(3x + 18 = 5x + 8\) (Underline)