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Look at the steps for solving the equation and choose a true statement.

3x - 9x + 1 = 2(-3x + 1) - 1
-6x + 1 = -6x + 2 - 1
-6x + 1 = -6x + 1
A. The only solution to the equation is -6
B. There are no solutions to the equation.
C. The only solution to the equation is 1
D. There is no way to know if there is a solution to the equation.
E. There are infinitely many solutions to the equation.

3 answers

To analyze the equation step by step and the results obtained:

  1. Starting with the equation:
    \( 3x - 9x + 1 = 2(-3x + 1) - 1 \)

  2. Simplifying the left side gives:
    \( -6x + 1 \)

  3. Simplifying the right side gives:
    \( 2(-3x + 1) - 1 = -6x + 2 - 1 = -6x + 1 \)

After simplification, we arrive at: \( -6x + 1 = -6x + 1 \)

Since both sides of the equation are identical, this means that the equation holds true for all values of \( x \). Therefore, there are infinitely many solutions to the equation.

Thus, the true statement is: E. There are infinitely many solutions to the equation.

are you sure

Yes, I am sure. Let's go through the reasoning step-by-step again to clarify:

Given the equation:

\[ 3x - 9x + 1 = 2(-3x + 1) - 1 \]

  1. Simplify the left-hand side: \[ 3x - 9x + 1 = -6x + 1 \]

  2. Simplify the right-hand side: \[ 2(-3x + 1) - 1 = -6x + 2 - 1 = -6x + 1 \]

Now we rewrite the full equation after simplification: \[ -6x + 1 = -6x + 1 \]

Both sides of the equation are exactly the same.

This indicates that the equation is true for all values of \( x \) because no matter what value is substituted for \( x \), both sides will always equal each other.

Therefore, the statement that holds true is: E. There are infinitely many solutions to the equation.

If you have any more questions or need further clarification, feel free to ask!