Let's go through each of the linear equations provided and determine whether they have no solution, one solution, or infinitely many solutions by solving them step by step.
1. Equation: \( 6x + 2 - 4x = 2(x + 2) \)
Step 1: Simplify both sides of the equation.
- Left Side: \( 6x - 4x + 2 = 2x + 2 \)
- Right Side: \( 2(x + 2) = 2x + 4 \)
Thus, the equation simplifies to: \[ 2x + 2 = 2x + 4 \]
Step 2: Isolate the variable. Subtract \( 2x \) from both sides: \[ 2 = 4 \]
Conclusion: Since \( 2 = 4 \) is a false statement, the equation has no solution.
2. Equation: \( 3(x + 3) = 7 + 3x + 6 - x \)
Step 1: Simplify both sides.
- Left Side: \( 3(x + 3) = 3x + 9 \)
- Right Side: \( 7 + 3x + 6 - x = 3x + 13 \)
Thus, the equation simplifies to: \[ 3x + 9 = 3x + 13 \]
Step 2: Isolate the variable. Subtract \( 3x \) from both sides: \[ 9 = 13 \]
Conclusion: Since \( 9 = 13 \) is a false statement, the equation has no solution.
3. Equation: \( 8(x + 2) = 5x + 16 + 3x \)
Step 1: Simplify both sides.
- Left Side: \( 8(x + 2) = 8x + 16 \)
- Right Side: \( 5x + 16 + 3x = 8x + 16 \)
Thus, the equation simplifies to: \[ 8x + 16 = 8x + 16 \]
Step 2: Isolate the variable. Subtract \( 8x \) from both sides: \[ 16 = 16 \]
Conclusion: Since \( 16 = 16 \) is a true statement, the equation has infinitely many solutions.
Summary of Results:
- Equation 1: No solution.
- Equation 2: No solution.
- Equation 3: Infinitely many solutions.