To determine which equations represent a linear function, we need to understand the definition of a linear function. A linear function is typically of the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] and [tex]\( b \)[/tex] are constants, and [tex]\( x \)[/tex] is the variable. Let's analyze each equation one by one.
1. Equation: [tex]\( x = 8 \)[/tex]
This equation represents a vertical line where [tex]\( x \)[/tex] is always equal to 8. It is not in the form of [tex]\( y = mx + b \)[/tex], hence it is not considered a linear function.
2. Equation: [tex]\( x - 3 = 5 \)[/tex]
Simplify this equation:
[tex]\[
x - 3 = 5 \quad \Rightarrow \quad x = 8
\][/tex]
This is essentially the same as the first equation, [tex]\( x = 8 \)[/tex], and similarly represents a vertical line. Therefore, it is not a linear function.
3. Equation: [tex]\( y = -4x^2 \)[/tex]
Here, the variable [tex]\( x \)[/tex] is squared. Any equation where the variable is raised to a power other than 1 does not represent a linear function. Thus, this is a quadratic function, not a linear one.
4. Equation: [tex]\( y = \frac{1}{4} x + 5 \)[/tex]
This equation is in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m = \frac{1}{4} \)[/tex] and [tex]\( b = 5 \)[/tex]. It matches the definition of a linear function.
Based on the analysis, the equation that represents a linear function is:
[tex]\[ y = \frac{1}{4} x + 5 \][/tex]
So, the indices of the linear equations in the provided list are:
[tex]\[ [3] \][/tex]
Therefore, the fourth equation [tex]\( y = \frac{1}{4} x + 5 \)[/tex] represents a linear function, corresponding to index 3 in the list.
