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 solution.
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!
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!
Answers
GPT-4o mini
Let's examine each equation to determine which ones have one solution, no solution, or infinitely many solutions.
1. **Equation:** \(2x + 3 = 2x + 7\)
- Subtract \(2x\) from both sides:
\[3 = 7\]
- This is a contradiction, so there is **no solution**.
2. **Equation:** \(7x + 3 = 7x + 3\)
- Both sides are identical, hence there are **infinitely many solutions**.
3. **Equation:** \(4x + 22 = 8x + 10\)
- Rearranging gives:
\[4x + 22 - 8x = 10\]
\[-4x + 22 = 10\]
\[-4x = -12\]
\[x = 3\]
- There is **one solution**: \(x = 3\).
4. **Equation:** \(5x + 5 = 5x + 4\)
- Subtract \(5x\) from both sides:
\[5 = 4\]
- This is a contradiction, so there is **no solution**.
5. **Equation:** \(3x + 18 = 5x + 8\)
- Rearranging gives:
\[3x + 18 - 5x = 8\]
\[-2x + 18 = 8\]
\[-2x = -10\]
\[x = 5\]
- There is **one solution**: \(x = 5\).
6. **Equation:** \(3(4x + 8) = 4(3x + 6)\)
- Expanding gives:
\[12x + 24 = 12x + 24\]
- Both sides are identical, hence there are **infinitely many solutions**.
Now, summarizing the findings:
- **Equations with infinitely many solutions:** \(7x + 3 = 7x + 3\) (Equation 2), \(3(4x + 8) = 4(3x + 6)\) (Equation 6)
- **Equations with no solution:** \(2x + 3 = 2x + 7\) (Equation 1), \(5x + 5 = 5x + 4\) (Equation 4)
- **Equations with one solution:** \(4x + 22 = 8x + 10\) (Equation 3), \(3x + 18 = 5x + 8\) (Equation 5)
**Step 1: Apply the required symbols to the equations**
- Circle around the equations with infinite solutions:
- \( \textcircled{2} \) \(7x + 3 = 7x + 3\)
- \( \textcircled{6} \) \(3(4x + 8) = 4(3x + 6)\)
- Square around the equations with no solution:
- \( \square \) \(2x + 3 = 2x + 7\)
- \( \square \) \(5x + 5 = 5x + 4\)
- Underline the equations with one solution:
- \( \underline{3} \) \(4x + 22 = 8x + 10\)
- \( \underline{5} \) \(3x + 18 = 5x + 8\)
**Step 2: Solve the equations with infinitely many solutions:**
**Equation:** \(7x + 3 = 7x + 3\) (This is true for all \(x\))
**Plug \(x = 3\):**
\[
7(3) + 3 = 7(3) + 3 \implies 21 + 3 = 21 + 3 \implies 24 = 24
\]
**Plug \(x = -2\):**
\[
7(-2) + 3 = 7(-2) + 3 \implies -14 + 3 = -14 + 3 \implies -11 = -11
\]
**Equation:** \(3(4x + 8) = 4(3x + 6)\)
**Plug \(x = 3\):**
\[
3(4(3) + 8) = 4(3(3) + 6) \implies 3(12 + 8) = 4(9 + 6) \implies 3(20) = 4(15) \implies 60 = 60
\]
**Plug \(x = -2\):**
\[
3(4(-2) + 8) = 4(3(-2) + 6) \implies 3(-8 + 8) = 4(-6 + 6) \implies 3(0) = 4(0) \implies 0 = 0
\]
**Both equations yield true statements for any chosen values, confirming infinite solutions!**
1. **Equation:** \(2x + 3 = 2x + 7\)
- Subtract \(2x\) from both sides:
\[3 = 7\]
- This is a contradiction, so there is **no solution**.
2. **Equation:** \(7x + 3 = 7x + 3\)
- Both sides are identical, hence there are **infinitely many solutions**.
3. **Equation:** \(4x + 22 = 8x + 10\)
- Rearranging gives:
\[4x + 22 - 8x = 10\]
\[-4x + 22 = 10\]
\[-4x = -12\]
\[x = 3\]
- There is **one solution**: \(x = 3\).
4. **Equation:** \(5x + 5 = 5x + 4\)
- Subtract \(5x\) from both sides:
\[5 = 4\]
- This is a contradiction, so there is **no solution**.
5. **Equation:** \(3x + 18 = 5x + 8\)
- Rearranging gives:
\[3x + 18 - 5x = 8\]
\[-2x + 18 = 8\]
\[-2x = -10\]
\[x = 5\]
- There is **one solution**: \(x = 5\).
6. **Equation:** \(3(4x + 8) = 4(3x + 6)\)
- Expanding gives:
\[12x + 24 = 12x + 24\]
- Both sides are identical, hence there are **infinitely many solutions**.
Now, summarizing the findings:
- **Equations with infinitely many solutions:** \(7x + 3 = 7x + 3\) (Equation 2), \(3(4x + 8) = 4(3x + 6)\) (Equation 6)
- **Equations with no solution:** \(2x + 3 = 2x + 7\) (Equation 1), \(5x + 5 = 5x + 4\) (Equation 4)
- **Equations with one solution:** \(4x + 22 = 8x + 10\) (Equation 3), \(3x + 18 = 5x + 8\) (Equation 5)
**Step 1: Apply the required symbols to the equations**
- Circle around the equations with infinite solutions:
- \( \textcircled{2} \) \(7x + 3 = 7x + 3\)
- \( \textcircled{6} \) \(3(4x + 8) = 4(3x + 6)\)
- Square around the equations with no solution:
- \( \square \) \(2x + 3 = 2x + 7\)
- \( \square \) \(5x + 5 = 5x + 4\)
- Underline the equations with one solution:
- \( \underline{3} \) \(4x + 22 = 8x + 10\)
- \( \underline{5} \) \(3x + 18 = 5x + 8\)
**Step 2: Solve the equations with infinitely many solutions:**
**Equation:** \(7x + 3 = 7x + 3\) (This is true for all \(x\))
**Plug \(x = 3\):**
\[
7(3) + 3 = 7(3) + 3 \implies 21 + 3 = 21 + 3 \implies 24 = 24
\]
**Plug \(x = -2\):**
\[
7(-2) + 3 = 7(-2) + 3 \implies -14 + 3 = -14 + 3 \implies -11 = -11
\]
**Equation:** \(3(4x + 8) = 4(3x + 6)\)
**Plug \(x = 3\):**
\[
3(4(3) + 8) = 4(3(3) + 6) \implies 3(12 + 8) = 4(9 + 6) \implies 3(20) = 4(15) \implies 60 = 60
\]
**Plug \(x = -2\):**
\[
3(4(-2) + 8) = 4(3(-2) + 6) \implies 3(-8 + 8) = 4(-6 + 6) \implies 3(0) = 4(0) \implies 0 = 0
\]
**Both equations yield true statements for any chosen values, confirming infinite solutions!**