1. Linear Equation with No Solution
Equation:
\[ 2x + 3 = 2x - 4 \]
Solving the Equation:
- Subtract \(2x\) from both sides: \[ 2x + 3 - 2x = 2x - 4 - 2x \] This simplifies to: \[ 3 = -4 \]
This statement is false, indicating that the original equation has no solution.
2. Linear Equation with One Solution
Equation:
\[ 3x - 5 = 4 \]
Solving the Equation:
-
Add \(5\) to both sides: \[ 3x - 5 + 5 = 4 + 5 \] This simplifies to: \[ 3x = 9 \]
-
Divide by \(3\): \[ x = \frac{9}{3} \] Thus, \[ x = 3 \]
This demonstrates that the equation has one solution, which is \(x = 3\).
3. Linear Equation with Infinitely Many Solutions
Equation:
\[ 4x + 2 = 2(2x + 1) \]
Solving the Equation:
-
Expand the right side: \[ 4x + 2 = 4x + 2 \]
-
Subtract \(4x\) from both sides: \[ 4x + 2 - 4x = 4x + 2 - 4x \] This simplifies to: \[ 2 = 2 \]
This statement is always true, regardless of the value of \(x\). Therefore the equation has infinitely many solutions, meaning any value of \(x\) satisfies the equation.