Questions: How many solutions does the system of equations below have? y=-5x-9 y=-2/5x-2/3 no solution one solution infinitely many solutions

Solution Steps

To determine the number of solutions for the system of linear equations, we need to check if the lines represented by the equations are parallel, coincident, or intersecting. This can be done by comparing their slopes and y-intercepts.

1. Extract the slopes and y-intercepts from both equations.
2. Compare the slopes:
   • If the slopes are equal and the y-intercepts are different, the lines are parallel and there is no solution.
   • If the slopes and y-intercepts are equal, the lines are coincident and there are infinitely many solutions.
   • If the slopes are different, the lines intersect at one point and there is one solution.

The given system of equations is: \[ y = -5x - 9 \] \[ y = \frac{-2}{5}x - \frac{2}{3} \]

From the first equation, we have:

• Slope \( m_1 = -5 \)
• Y-intercept \( b_1 = -9 \)

From the second equation, we have:

• Slope \( m_2 = -0.4 \) (which is equivalent to \( \frac{-2}{5} \))
• Y-intercept \( b_2 = -0.6667 \) (which is equivalent to \( \frac{-2}{3} \))

We compare the slopes:

• \( m_1 = -5 \)
• \( m_2 = -0.4 \)

Since \( m_1 \neq m_2 \), the lines are not parallel.

Since the slopes are different, the lines intersect at one point. Therefore, the system of equations has one solution.

Final Answer