Question

Directions: Use this worksheet to record your answers to the questions about linear
equations. When you are finished, save this worksheet with your answers and submit
it for a portfolio grade.
Investigate and explain the different number of solution an equation can have.

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

To analyze the equations for their number of solutions, let’s check each equation one by one:

1. **Equation: 2x + 3 = 2x + 7**
- Subtract 2x from both sides: 3 = 7
- This is a false statement.
- **No solution.** (Square)

2. **Equation: 7x + 3 = 7x + 3**
- Both sides are identical.
- This is a true statement for all values of x.
- **Infinitely many solutions.** (Circle)

3. **Equation: 4x + 22 = 8x + 10**
- Subtract 4x from both sides: 22 = 4x + 10
- Subtract 10 from both sides: 12 = 4x
- Divide by 4: x = 3
- **One solution.** (Underline)

4. **Equation: 5x + 5 = 5x + 4**
- Subtract 5x from both sides: 5 = 4
- This is a false statement.
- **No solution.** (Square)

5. **Equation: 3x + 18 = 5x + 8**
- Subtract 3x from both sides: 18 = 2x + 8
- Subtract 8 from both sides: 10 = 2x
- Divide by 2: x = 5
- **One solution.** (Underline)

6. **Equation: 3(4x + 8) = 4(3x + 6)**
- Expand both sides: 12x + 24 = 12x + 24
- Both sides are identical.
- This is a true statement for all values of x.
- **Infinitely many solutions.** (Circle)

Here is the classification of the equations:

- **Infinitely many solutions**:
- 7x + 3 = 7x + 3 (●)
- 3(4x + 8) = 4(3x + 6) (●)

- **No solution**:
- 2x + 3 = 2x + 7 (□)
- 5x + 5 = 5x + 4 (□)

- **One solution**:
- 4x + 22 = 8x + 10 (____)
- 3x + 18 = 5x + 8 (____)

Thus, your final answers should look like this:

- **Infinitely many solutions** (circled):
- 7x + 3 = 7x + 3
- 3(4x + 8) = 4(3x + 6)

- **No solution** (squared):
- 2x + 3 = 2x + 7
- 5x + 5 = 5x + 4

- **One solution** (underlined):
- 4x + 22 = 8x + 10
- 3x + 18 = 5x + 8