Create a single variable linear equation that has no solution. Solve the equation algebraically to prove that it does not have a solution.

Create a single variable linear equation that has one solution. Solve the equation algebraically to prove that there is one distinct solution for the equation.
Create a single variable linear equation that has infinitely many solutions. Solve the equation algebraically to prove that there is an infinite number of solutions for the equation

5 answers

1. Linear Equation with No Solution

Equation:
\[ 2x + 3 = 2x - 4 \]

Solving the Equation:

  1. 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:

  1. Add \(5\) to both sides: \[ 3x - 5 + 5 = 4 + 5 \] This simplifies to: \[ 3x = 9 \]

  2. 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:

  1. Expand the right side: \[ 4x + 2 = 4x + 2 \]

  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.

can you make it less advanced

Sure! Let's simplify the explanations so they're more straightforward.

1. Linear Equation with No Solution

Equation:
\[ 2x + 3 = 2x - 4 \]

Solving the Equation:

  1. Start with the equation: \[ 2x + 3 = 2x - 4 \]

  2. Subtract \( 2x \) from both sides: \[ 3 = -4 \]

This is clearly not true (3 does not equal -4). Since we ended up with a false statement, this equation has no solution.


2. Linear Equation with One Solution

Equation:
\[ 3x - 5 = 4 \]

Solving the Equation:

  1. Start with the equation: \[ 3x - 5 = 4 \]

  2. Add \( 5 \) to both sides: \[ 3x = 9 \]

  3. Now, divide by \( 3 \): \[ x = 3 \]

So, the equation has one solution, which is \( x = 3 \).


3. Linear Equation with Infinitely Many Solutions

Equation:
\[ 4x + 2 = 4x + 2 \]

Solving the Equation:

  1. Start with the equation: \[ 4x + 2 = 4x + 2 \]

  2. Notice that both sides are exactly the same.

This means that the statement is always true—no matter what value \( x \) takes. Therefore, this equation has infinitely many solutions because any value for \( x \) will work!

can you use less words

Of course! Here are the simplified versions with fewer words:

1. Linear Equation with No Solution

Equation:
\[ 2x + 3 = 2x - 4 \]

Solution:

  1. Subtract \( 2x \): \[ 3 = -4 \] (false)

Conclusion: No solution.


2. Linear Equation with One Solution

Equation:
\[ 3x - 5 = 4 \]

Solution:

  1. Add \( 5 \): \[ 3x = 9 \]

  2. Divide by \( 3 \): \[ x = 3 \]

Conclusion: One solution: \( x = 3 \).


3. Linear Equation with Infinitely Many Solutions

Equation:
\[ 4x + 2 = 4x + 2 \]

Solution: Both sides are the same.

Conclusion: Infinitely many solutions. Any \( x \) works!