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

Question

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

Answer 1

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.

Answer 2

can you make it less advanced

Answer 3

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.

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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 \( x = 3 \).

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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!

Answer 4

can you use less words

Answer 5

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.

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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: \( x = 3 \).

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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!