13. Isabella begins riding a bike on a trail at mile 6 at a rate of \( 8 \mathrm{mi} / \mathrm{h} \). Tanaysha beings riding a bike from the beginning of the trail at a rate of \( 9 \mathrm{mi} / \mathrm{h} \). A. Write a system of linear equations to model the situation. B. Identify the system as independent, dependent, or inconsistent. Justify your reasoning.

**A. Constructing the System of Linear Equations** Let \( t \) be the time (in hours) after they begin riding. - **Isabella’s distance:** Isabella starts at mile 6 and rides at \( 8 \) miles per hour. Her distance \( d_I \) (in miles) along the trail after time \( t \) is given by \[ d_I = 6 + 8t. \] - **Tanaysha’s distance:** Tanaysha starts at mile 0 and rides at \( 9 \) miles per hour. Her distance \( d_T \) (in miles) after time \( t \) is given by \[ d_T = 9t. \] If we want to find when they meet at the same location on the trail, set their distances equal: \[ 6 + 8t = 9t. \] Thus, the system of equations modeling the situation is: \[ \begin{cases} d_I = 6 + 8t, \\ d_T = 9t, \\ 6 + 8t = 9t. \end{cases} \] **B. Classification of the System** The equations \( d_I = 6 + 8t \) and \( d_T = 9t \) represent two distinct linear relationships. When set equal, the equation \[ 6 + 8t = 9t \] simplifies to \[ 6 = t, \] which gives a unique solution for \( t \). Since the two equations represent two distinct lines that intersect at exactly one point (they have different slopes), the system is **independent**. **Justification:** - The line representing Isabella’s motion has a slope of \( 8 \) and a \( y \)-intercept of \( 6 \). - The line representing Tanaysha’s motion has a slope of \( 9 \) and a \( y \)-intercept of \( 0 \). Since the slopes are different, the lines intersect at one unique point, hence the system is independent.