Let's carefully analyze the given equation and the steps taken by both Roy and Sam:
The given equation is:
[tex]\[ 3x + 4 = 5x - 6 \][/tex]
### Roy's Work:
Roy starts with:
[tex]\[ -3x + 4 = 5x - 6 \][/tex]
Immediately, there is an apparent issue here because the original equation was [tex]\( 3x + 4 = 5x - 6 \)[/tex], and somehow Roy has made [tex]\( -3x + 4 \)[/tex] from it, which is incorrect. Therefore, Roy's work is incorrect from the start.
### Sam's Work:
Sam starts with the original equation:
[tex]\[ 3x + 4 = 5x - 6 \][/tex]
Next, he isolates the [tex]\( x \)[/tex]-terms on one side by subtracting [tex]\( 3x \)[/tex] from both sides and adding 6 to both sides:
[tex]\[ 4 = 2x - 6 \][/tex]
We need to verify if this transformation is consistent with algebraic principles. Let's manually verify Sam's steps:
1. Start with the original equation:
[tex]\[ 3x + 4 = 5x - 6 \][/tex]
2. Subtract [tex]\( 3x \)[/tex] from both sides:
[tex]\[ 4 = 5x - 3x - 6 \][/tex]
[tex]\[ 4 = 2x - 6 \][/tex]
We can see that Sam has made the correct steps by using the addition property of equality. This method keeps the equation balanced by ensuring both sides are adjusted similarly.
Therefore, the correct answer is:
C. Sam's work is correct so far. He used the addition property of equality.
