How many solutions does this equation have?7(2w+1)=14w+1no solutionone solutiontwo solutionsinfinitely many solutionsSubmit

Break Down the Problem

1. Identify the equation: The equation given is $7(2w + 1) = 14w + 1$.
2. Simplify both sides: We need to expand the left side and simplify the equation.

Relevant Concepts

• Equality of equations: Two expressions are equal if they yield the same value for the same variable.
• Solving linear equations: Isolate the variable on one side to determine the solution.

Analysis and Detail

1. Start by expanding the left-hand side: $7(2w + 1) = 14w + 7$ So the equation becomes: $14w + 7 = 14w + 1$

2. Next, subtract $14w$ from both sides: $14w + 7 - 14w = 14w + 1 - 14w$ This simplifies to: $7 = 1$

3. The equation $7 = 1$ is a contradiction. Therefore, this suggests that there are no values of $w$ that satisfy the equation.

Verify and Summarize

1. Since we reached a contradiction, it verifies that the original equation does not hold true for any value of $w$.
2. Thus, there are no solutions to the equation.

Final Answer

The equation $7(2w + 1) = 14w + 1$ has no solution.