Let's analyze each of the given expressions to determine which of them is a linear equation in two variables.
a. [tex]\( 3p - 6.7 \)[/tex]
This is not an equation; it's just an expression. For it to be considered an equation, it must have an equality sign ("=") and typically two variables in the standard linear equation form [tex]\( ax + by = c \)[/tex].
b. [tex]\( 3a = 5 - b \)[/tex]
This is an equation. To see if it fits the form of a linear equation in two variables [tex]\( ax + by = c \)[/tex], we can rewrite it:
[tex]\[
3a = 5 - b
\][/tex]
By adding [tex]\( b \)[/tex] to both sides to get all variable terms on one side of the equation:
[tex]\[
3a + b = 5
\][/tex]
This is now in the form [tex]\( ax + by = c \)[/tex], where [tex]\( a = 3 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = 5 \)[/tex]. Therefore, this is a linear equation in two variables.
c. [tex]\( 2P - 7a = -3P \)[/tex]
This is also an equation. To determine if it fits the standard form of a linear equation in two variables [tex]\( ax + by = c \)[/tex]:
[tex]\[
2P - 7a = -3P
\][/tex]
Add [tex]\( 3P \)[/tex] to both sides to get all [tex]\( P \)[/tex] terms on one side:
[tex]\[
2P + 3P - 7a = 0
\][/tex]
Combine like terms:
[tex]\[
5P - 7a = 0
\][/tex]
Now, this is in the form [tex]\( ax + by = c \)[/tex], where [tex]\( a = 5 \)[/tex], [tex]\( b = -7 \)[/tex], and [tex]\( c = 0 \)[/tex]. Therefore, this is a linear equation in two variables.
d. [tex]\( -n \cdot 3 \cdot 4n \)[/tex]
This is an expression involving multiplication of variables and constants, not an equation. It can be simplified to [tex]\( -12n^2 \)[/tex], but it still lacks an equality sign and doesn't conform to the linear equation form of [tex]\( ax + by = c \)[/tex].
e. [tex]\( \frac{x}{0} + \frac{2y}{5}: 10 \)[/tex]
Here, we encounter a term [tex]\( \frac{x}{0} \)[/tex]. Division by zero is undefined in mathematics, making this an invalid expression. Thus, it cannot be considered a linear equation in two variables.
Given the above analyses, the correct options that represent linear equations in two variables are:
b. [tex]\( 3a = 5 - b \)[/tex]
c. [tex]\( 2P - 7a = -3P \)[/tex]
So, the correct choices for linear equations in two variables are:
b. [tex]\( 3a = 5 - b \)[/tex]
c. [tex]\( 2P - 7a = -3P \)[/tex]
