Let’s solve each of the equations step by step and determine how many solutions each has.
1. Equation: \( 2x + 3 = 2x + 7 \)
Step 1: Subtract \( 2x \) from both sides: \[ 2x + 3 - 2x = 2x + 7 - 2x \ 3 = 7 \] This is not true (3 does not equal 7).
Conclusion: There are None solutions.
2. Equation: \( 7x + 3 = 7x + 3 \)
Step 1: Subtract \( 7x \) from both sides: \[ 7x + 3 - 7x = 7x + 3 - 7x \ 3 = 3 \] This is always true.
Conclusion: There are Infinite solutions.
3. Equation: \( 4x + 22 = 8x + 10 \)
Step 1: Subtract \( 4x \) from both sides: \[ 4x + 22 - 4x = 8x + 10 - 4x \ 22 = 4x + 10 \]
Step 2: Subtract 10 from both sides: \[ 22 - 10 = 4x \ 12 = 4x \]
Step 3: Divide both sides by 4: \[ x = 3 \]
Conclusion: There is One solution.
4. Equation: \( 5x + 5 = 5x + 4 \)
Step 1: Subtract \( 5x \) from both sides: \[ 5x + 5 - 5x = 5x + 4 - 5x \ 5 = 4 \] This is not true (5 does not equal 4).
Conclusion: There are None solutions.
5. Equation: \( 3x + 18 = 5x + 8 \)
Step 1: Subtract \( 3x \) from both sides: \[ 3x + 18 - 3x = 5x + 8 - 3x \ 18 = 2x + 8 \]
Step 2: Subtract 8 from both sides: \[ 18 - 8 = 2x \ 10 = 2x \]
Step 3: Divide both sides by 2: \[ x = 5 \]
Conclusion: There is One solution.
6. Equation: \( 3(4x + 8) = 4(3x + 6) \)
Step 1: Distribute both sides: \[ 12x + 24 = 12x + 24 \]
Step 2: Subtract \( 12x \) from both sides: \[ 12x + 24 - 12x = 12x + 24 - 12x \ 24 = 24 \] This is always true.
Conclusion: There are Infinite solutions.
Summary of Solutions
- \( 2x + 3 = 2x + 7 \) - None
- \( 7x + 3 = 7x + 3 \) - Infinite
- \( 4x + 22 = 8x + 10 \) - One
- \( 5x + 5 = 5x + 4 \) - None
- \( 3x + 18 = 5x + 8 \) - One
- \( 3(4x + 8) = 4(3x + 6) \) - Infinite