Question
1 Examine the six different equations that follow: 1. 2x+3=2x+7 2. 7x+3=7x+3 3. 4x+22=8x+10 4. 5x+5=5x+4 5. 3x+18=5x+8 6. 3(4x+8)=4(3x+6) 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 solutio
Answers
GPT-4o mini
Let's analyze each of the six equations to determine if they have one solution, no solution, or infinitely many solutions.
1. **2x + 3 = 2x + 7**
- Subtracting \(2x\) from both sides gives us \(3 = 7\), which is false.
- **No solution.** (Put a square around this equation)
2. **7x + 3 = 7x + 3**
- Both sides are equal, which is always true regardless of the value of \(x\).
- **Infinitely many solutions.** (Put a circle around this equation)
3. **4x + 22 = 8x + 10**
- Rearranging gives \(4x = 12 \Rightarrow x = 3\).
- **One solution.** (Underline this equation)
4. **5x + 5 = 5x + 4**
- Subtracting \(5x\) from both sides gives \(5 = 4\), which is false.
- **No solution.** (Put a square around this equation)
5. **3x + 18 = 5x + 8**
- Rearranging gives \(-2x = -10 \Rightarrow x = 5\).
- **One solution.** (Underline this equation)
6. **3(4x + 8) = 4(3x + 6)**
- Expanding both sides gives \(12x + 24 = 12x + 24\).
- Both sides are equal, which is always true.
- **Infinitely many solutions.** (Put a circle around this equation)
### Summary:
- **Infinitely many solutions:** 2, 6 (Put a circle around these)
- **No solution:** 1, 4 (Put a square around these)
- **One solution:** 3, 5 (Underline these)
Here’s how it looks:
1. **No Solution:**
- 1. **2x + 3 = 2x + 7** (Square)
- 4. **5x + 5 = 5x + 4** (Square)
2. **Infinitely Many Solutions:**
- 2. **7x + 3 = 7x + 3** (Circle)
- 6. **3(4x + 8) = 4(3x + 6)** (Circle)
3. **One Solution:**
- 3. **4x + 22 = 8x + 10** (Underline)
- 5. **3x + 18 = 5x + 8** (Underline)
1. **2x + 3 = 2x + 7**
- Subtracting \(2x\) from both sides gives us \(3 = 7\), which is false.
- **No solution.** (Put a square around this equation)
2. **7x + 3 = 7x + 3**
- Both sides are equal, which is always true regardless of the value of \(x\).
- **Infinitely many solutions.** (Put a circle around this equation)
3. **4x + 22 = 8x + 10**
- Rearranging gives \(4x = 12 \Rightarrow x = 3\).
- **One solution.** (Underline this equation)
4. **5x + 5 = 5x + 4**
- Subtracting \(5x\) from both sides gives \(5 = 4\), which is false.
- **No solution.** (Put a square around this equation)
5. **3x + 18 = 5x + 8**
- Rearranging gives \(-2x = -10 \Rightarrow x = 5\).
- **One solution.** (Underline this equation)
6. **3(4x + 8) = 4(3x + 6)**
- Expanding both sides gives \(12x + 24 = 12x + 24\).
- Both sides are equal, which is always true.
- **Infinitely many solutions.** (Put a circle around this equation)
### Summary:
- **Infinitely many solutions:** 2, 6 (Put a circle around these)
- **No solution:** 1, 4 (Put a square around these)
- **One solution:** 3, 5 (Underline these)
Here’s how it looks:
1. **No Solution:**
- 1. **2x + 3 = 2x + 7** (Square)
- 4. **5x + 5 = 5x + 4** (Square)
2. **Infinitely Many Solutions:**
- 2. **7x + 3 = 7x + 3** (Circle)
- 6. **3(4x + 8) = 4(3x + 6)** (Circle)
3. **One Solution:**
- 3. **4x + 22 = 8x + 10** (Underline)
- 5. **3x + 18 = 5x + 8** (Underline)
2 Solve the two equations above that have infinite solutions? Show your work! Plug x=3 into each equation and solve. Show your work! Plug x=-2 into each equation and solve. Show your work!