To solve the equation [tex]\(2a + 3b - 4a = a - b + 5a\)[/tex] using only the commutative property in the first step, we need to rearrange the terms on both sides of the equation without changing their order fundamentally, i.e., we'll just group and move terms but won't combine or simplify them.
Let's analyze the possible expressions step-by-step:
1. (Expression: [tex]\(2a - 4a + 3b = a + 5a - b\)[/tex])
- On the left-hand side, we simply rearrange [tex]\(2a\)[/tex] and [tex]\(-4a\)[/tex], and [tex]\(3b\)[/tex] remains in the same place.
- On the right-hand side, we reorder [tex]\(a\)[/tex] and [tex]\(5a\)[/tex], while [tex]\(-b\)[/tex] stays with [tex]\(5a\)[/tex].
- This is a possible valid expression.
2. (Expression: [tex]\(-2a + 3b = a - b + 5a\)[/tex])
- Here, we simplify [tex]\(2a - 4a\)[/tex] to [tex]\(-2a\)[/tex] and keep [tex]\(3b\)[/tex] as it is.
- On the right-hand side, we don't change the arrangement: [tex]\(a - b + 5a\)[/tex].
- This does not involve using the commutative property correctly for the first step as it simplifies the terms. Therefore, this is not valid.
3. (Expression: [tex]\(-2a + 3b = 6a - b\)[/tex])
- On the left-hand side, combining [tex]\(2a - 4a\)[/tex] as [tex]\(-2a\)[/tex], and keeping [tex]\(3b\)[/tex].
- The right-hand side simplifies [tex]\(a + 5a\)[/tex] to [tex]\(6a - b\)[/tex].
- This involves simplification, not just rearrangement, so this is invalid.
4. (Expression: [tex]\(2a - 4a + 3b = a - b + 5a\)[/tex])
- Here, the left-hand side is rearranged as [tex]\(2a - 4a + 3b\)[/tex].
- The right-hand side is rearranged into [tex]\(a - b + 5a\)[/tex].
- This is the same as the first one, just written differently.
5. (Expression: [tex]\(2a + 3b - 4a = a + 5a - b\)[/tex])
- This is the original expression but rewritten with the terms moved around.
- This accurately represents a commutatively modified version.
Hence, the correct expressions that Alec could get using only the commutative property for the first step are:
[tex]\(2a - 4a + 3b = a + 5a - b\)[/tex]
[tex]\(2a - 4a + 3b = a - b + 5a\)[/tex]
* [tex]\(2a + 3b - 4a = a + 5a - b\)[/tex]
The valid ones among the given options are:
- [tex]\(2a - 4a + 3b = a + 5a - b\)[/tex]
- [tex]\(-2a + 3b = a - b + 5a\)[/tex]
- [tex]\(2a - 4a + 3b = a - b + 5a\)[/tex]
- [tex]\(2a + 3b - 4a = a + 5a - b\)[/tex]
So we should correctly select all valid expressions.
