Question

Tucker was asked to solve the equation 5x + 3 = 6x + 1. He did not know if his first step should be to add 5 negative x-tiles, or 1 negative unit tile, to both sides. What advice would you give Tucker to help him decide on his first step? Explain.

Answer 1

To help Tucker decide on his first step in solving the equation \( 5x + 3 = 6x + 1 \), we can analyze the equation and determine which step would be most effective.

The goal in solving for \( x \) is to isolate the variable on one side of the equation. In this case, Tucker has two options for the first step:

1. **Add 5 negative \( x \)-tiles (which would essentially mean moving \( 6x \) from the right side to the left side)**: This would lead to rearranging the equation as follows:
\[
5x + 3 - 6x = 1
\]
Simplifying this would give:
\[
-x + 3 = 1
\]
This is a valid step and can lead to isolating \( x \).

2. **Add 1 negative unit tile (which means subtracting 1 from both sides)**: This would lead to:
\[
5x + 3 - 1 = 6x + 1 - 1
\]
Simplifying this would give:
\[
5x + 2 = 6x
\]
This is also a valid step and can help isolate \( x \).

Both steps are valid, but one might be simpler or more intuitive depending on Tucker's understanding of the equation.

**Advice**:

It might be easier for Tucker to start by moving the \( 6x \) term to the left side by adding \( -6x \) (or "5 negative \( x \)-tiles") to both sides. This approach reduces the number of terms associated with \( x \) right away, potentially making the next steps more straightforward.

After moving \( 6x \) to the left, he can easily combine like terms and continue to solve for \( x \).

So, I would suggest that Tucker adds 5 negative \( x \)-tiles to both sides to start. It usually helps to eliminate the variable term on one side as a first step.