Question

Make up an equation that has no solution and give it to a fellow student to solve. Ask the fellow student to write a critique of your equation.

Short answer

The equation \( 2x + 3 = 2x + 5 \) has no solution because it simplifies to the contradiction \( 3 = 5 \).

Step-by-step solution

- Formulate an Equation

Create an equation where the variables and coefficients lead to a contradiction. An example of such an equation could be: \[2x + 3 = 2x + 5\]

- Isolate the Variables

Attempt to isolate the variable on one side. Subtract \(2x\) from both sides of the equation: \[2x + 3 - 2x = 2x + 5 - 2x\] This simplifies to: \[3 = 5\]

- Identify the Contradiction

Notice that the simplified result is a contradiction since 3 can never be equal to 5. This indicates that the original equation has no solution.

Key concepts

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Understanding algebraic equations is crucial for solving math problems. An algebraic equation is a mathematical statement that shows the equality of two expressions with variables and constants. For example, in the equation 2x + 3 = 2x + 5 , the values of x are unknown. The goal is to find out if there is a value of x that makes this equation true. We use different methods to manipulate the equation and find the value of the variable.

However, not all algebraic equations have solutions. Some equations may lead to contradictions, meaning there is no possible value of the variable that can satisfy the equation. Let's explore this concept further.

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A contradiction in an equation occurs when the simplified form of the equation results in an impossible statement. For example, let's reconsider our equation:

2x + 3 = 2x + 5. When trying to isolate the variable x, we subtract 2x from both sides:

2x + 3 - 2x = 2x + 5 - 2x .

This simplifies to: 3 = 5. Notice that 3 can never equal 5, which is a clear contradiction. Thus, this equation has no solution.

In mathematical terms, a contradiction occurs when an equation simplifies to a false statement. This means that no value of the unknown variable will make the equation true, so the original equation has no solution.

• Contradictions often emerge when similar terms cancel each other out.
• Always look for simplifications that produce false statements.
• Contradictions reveal that an equation can't be satisfied by any value.

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Solving equations involves finding the value of the unknown variable that makes the equation true. There are different strategies for solving equations, but sometimes, the process reveals that no solution exists. Let's revisit our example to understand the steps involved:

2x + 3 = 2x + 5.

Step-by-Step Guide:

• Step 1 - Formulate the Equation: Create or identify an equation. Here, we use 2x + 3 = 2x + 5.
• Step 2 - Isolate the Variable: Subtract 2x from both sides to move the variable terms to one side:
  2x + 3 - 2x = 2x + 5 - 2x
• This simplifies to 3 = 5.
• Step 3 - Identify Contradiction: Notice that the equation simplifies to a false statement.
  3 can never equal 5, showing that there is no solution for the equation.

Always check for contradictions when solving equations. When an equation simplifies to a contradiction, it indicates that there's no value for the variable that satisfies the original equation.

• Remember to perform the same operation on both sides of the equation.
• Simplify step-by-step to reveal possible contradictions.
• Recognize that not all equations are solvable. Identifying contradictions is part of mastering algebra.

With practice, solving equations and identifying contradictions will become more natural. Keep studying and practicing!