Let's analyze Bernard's steps in solving the equation [tex]\(5x + (-4) = 6x + 4\)[/tex].
Bernard's goal is to isolate the variable [tex]\(x\)[/tex] on one side of the equation. Here is a step-by-step explanation of his process:
1. The given equation is [tex]\(5x + (-4) = 6x + 4\)[/tex].
2. Bernard adds 5 negative [tex]\(x\)[/tex]-tiles to both sides of the equation to eliminate the [tex]\(x\)[/tex] term on the left side:
[tex]\[
(5x + (-4)) + (-5x) = (6x + 4) + (-5x)
\][/tex]
3. Simplifying both sides:
- On the left side: [tex]\(5x + (-5x) + (-4)\)[/tex]
- On the right side: [tex]\(6x + (-5x) + 4\)[/tex]
This results in:
[tex]\[
-4 = x + 4
\][/tex]
4. Bernard now has [tex]\(-4\)[/tex] on the left side and [tex]\(x + 4\)[/tex] on the right side.
From this manipulation, it's clear that Bernard's purpose was to create zero pairs (pairs of terms that sum to zero) on the left side of the equation to isolate the constant term [tex]\(-4\)[/tex]. This allows him to express [tex]\(x\)[/tex] more simply on the right side without the complication of additional [tex]\(x\)[/tex] terms.
So, the correct explanation for why Bernard added 5 negative [tex]\(x\)[/tex] tiles to both sides is:
He wanted to create zero pairs on the left side of the equation to get a positive coefficient of [tex]\(x\)[/tex] on the right side.
