Let's examine each given expression to determine which ones are linear equations.
### Expression (a): [tex]\( x^2 + x = 2 \)[/tex]
To determine whether this equation is linear, we need to look at the highest power of the variable [tex]\( x \)[/tex].
- The term [tex]\( x^2 \)[/tex] has a degree of 2.
- The term [tex]\( x \)[/tex] has a degree of 1.
Since the highest degree of [tex]\( x \)[/tex] in this expression is 2, this equation is not linear. Linear equations must have variables raised only to the first power.
### Expression (b): [tex]\( 3x + 5 = 11 \)[/tex]
Next, we examine this equation:
- The term [tex]\( 3x \)[/tex] has a degree of 1.
- The term [tex]\( 5 \)[/tex] is a constant (it has [tex]\( x^0 \)[/tex], which is considered of degree 0).
Since the highest degree of [tex]\( x \)[/tex] in this expression is 1, this equation is linear. Linear equations are of the form [tex]\( ax + b = c \)[/tex], where the variable [tex]\( x \)[/tex] is raised to the power of 1.
### Expression (c): [tex]\( 5 + 7 = 12 \)[/tex]
Lastly, we evaluate this expression:
- The terms [tex]\( 5 \)[/tex] and [tex]\( 7 \)[/tex] are constants and do not contain the variable [tex]\( x \)[/tex].
This expression does not have a variable term at all, meaning it is essentially a statement about constants. Therefore, it can be considered as a linear equation in a trivial way with the variable [tex]\( x \)[/tex] absent (degree [tex]\( x^0 \)[/tex]).
Since identifying linear equations typically requires the presence of a variable raised to the first power, and considering that [tex]\( 5 + 7 = 12 \)[/tex] is simply an arithmetic statement, this doesn't fit the standard form of a linear equation involving a variable.
### Conclusion
The linear equation from the given expressions is:
- (b) [tex]\( 3x + 5 = 11 \)[/tex]
Therefore, the indices of the linear equations are:
- (b)
Thus, the linear equation identified is: (b) [tex]\( 3x + 5 = 11 \)[/tex].
