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Solve for x to determine if the equation has one, none, or infinite solutions.

5(2x + 2) - 7x = 3x + 6
A. The equation has infinite solutions.
B. The equation has no solution.
C. The equation has one solution: x = 6.
D. The equation has one solution: x = 10.

3 answers

To solve the equation \( 5(2x + 2) - 7x = 3x + 6 \), let's simplify it step by step.

  1. Distribute \( 5 \) in the left side: \[ 10x + 10 - 7x = 3x + 6 \]

  2. Combine like terms on the left side: \[ (10x - 7x) + 10 = 3x + 6 \] \[ 3x + 10 = 3x + 6 \]

  3. Next, subtract \( 3x \) from both sides: \[ 10 = 6 \]

This statement is false, indicating that there are no values of \( x \) that satisfy the equation.

Since we ended up with a false statement, this means that the original equation has no solution.

Therefore, the answer is: B. The equation has no solution.

are you sure this is correct

Let's double-check the work step-by-step to ensure accuracy.

We start with the original equation:

\[ 5(2x + 2) - 7x = 3x + 6 \]

  1. Distributing \( 5 \) on the left side:

\[ 5 \cdot 2x + 5 \cdot 2 - 7x = 3x + 6 \]

This simplifies to:

\[ 10x + 10 - 7x = 3x + 6 \]

  1. Combine like terms on the left side:

\[ (10x - 7x) + 10 = 3x + 6 \] \[ 3x + 10 = 3x + 6 \]

  1. Now, let’s isolate the variable by subtracting \( 3x \) from both sides:

\[ 10 = 6 \]

This results in a false statement (10 is not equal to 6). Therefore, the equation has no solutions.

So the conclusion is indeed correct. The original equation has:

B. The equation has no solution.

Thank you for prompting the verification!