To determine which expressions are equivalent to the expression [tex]\( 4a - 6b + 3c \)[/tex], let's simplify each of the given expressions step by step.
### Expression 1: [tex]\( a + 3(a - 2b + 3c) \)[/tex]
First, we simplify inside the parentheses:
[tex]\[ 3(a - 2b + 3c) = 3a - 6b + 9c \][/tex]
Now add [tex]\( a \)[/tex]:
[tex]\[ a + 3a - 6b + 9c = 4a - 6b + 9c \][/tex]
This expression is:
[tex]\[ 4a - 6b + 9c \][/tex]
This is not equivalent to [tex]\( 4a - 6b + 3c \)[/tex].
### Expression 2: [tex]\( 4a + 3(2b + c) \)[/tex]
First, we simplify inside the parentheses:
[tex]\[ 3(2b + c) = 6b + 3c \][/tex]
Now add [tex]\( 4a \)[/tex]:
[tex]\[ 4a + 6b + 3c = 4a + 6b + 3c \][/tex]
This expression is:
[tex]\[ 4a + 6b + 3c \][/tex]
This is also not equivalent to [tex]\( 4a - 6b + 3c \)[/tex].
### Expression 3: [tex]\( 2(2a - 3b + c) + c \)[/tex]
First, we distribute 2 inside the parentheses:
[tex]\[ 2(2a - 3b + c) = 4a - 6b + 2c \][/tex]
Now add [tex]\( c \)[/tex]:
[tex]\[ 4a - 6b + 2c + c = 4a - 6b + 3c \][/tex]
This expression is:
[tex]\[ 4a - 6b + 3c \][/tex]
This is equivalent to [tex]\( 4a - 6b + 3c \)[/tex].
### Expression 4: [tex]\( 2(2a - 3b) + 3c \)[/tex]
First, we distribute 2 inside the parentheses:
[tex]\[ 2(2a - 3b) = 4a - 6b \][/tex]
Now add [tex]\( 3c \)[/tex]:
[tex]\[ 4a - 6b + 3c = 4a - 6b + 3c \][/tex]
This expression is:
[tex]\[ 4a - 6b + 3c \][/tex]
This is equivalent to [tex]\( 4a - 6b + 3c \)[/tex].
### Summary
The expressions equivalent to [tex]\( 4a - 6b + 3c \)[/tex] are:
[tex]\[ 2(2a - 3b + c) + c \][/tex]
[tex]\[ 2(2a - 3b) + 3c \][/tex]
