Determine whether the following system has one solution, no solution, or infinitely many solutions. Show your work. y = -x/3 + 5 y = (-1/5)x - 2

Steps to Solve

1. Compare the slopes and y-intercepts

   The given equations are in slope-intercept form, $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept

   Equation 1: $y = -\frac{1}{3}x + 5$ slope $m_1 = -\frac{1}{3}$ and y-intercept $b_1 = 5$

   Equation 2: $y = -\frac{1}{5}x - 2$ slope $m_2 = -\frac{1}{5}$ and y-intercept $b_2 = -2$

2. Determine the number of solutions based on slopes and y-intercepts

   • If the slopes are different ($m_1 \neq m_2$), the system has one solution.
   • If the slopes are the same ($m_1 = m_2$) and the y-intercepts are different ($b_1 \neq b_2$), the system has no solution (parallel lines).
   • If the slopes are the same ($m_1 = m_2$) and the y-intercepts are the same ($b_1 = b_2$), the system has infinitely many solutions (same line).

3. Apply the conditions to our equations

   We have $m_1 = -\frac{1}{3}$ and $m_2 = -\frac{1}{5}$. Since $-\frac{1}{3} \neq -\frac{1}{5}$, the slopes are different.

   Therefore, since the slopes are different, the system has one solution.