To analyze the equation \( y = \frac{6}{x} - 3 \) to determine if it is linear, we can examine its form.
A linear equation can be written in the standard form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
The given equation \( y = \frac{6}{x} - 3 \) includes the term \( \frac{6}{x} \), which can also be written as \( 6x^{-1} \). This term indicates that the relationship between \( x \) and \( y \) is not linear because the variable \( x \) is in the denominator (or equivalently, has a negative exponent).
Since it cannot be expressed in the form \( y = mx + b \), the given equation is not linear. Instead, it represents a hyperbola.