Question

How many solutions are there to the following system of equations? Equation 1: The line that passes through the points (0,0) and (2,3) Equation 2: The line that passes through the points (2,3) and (4,6) One Solution No Solutions Infinitely Many Solutions

Answer 1

Equation 1:

Let:

`\begin{gathered} (x1,y1)=(0,0) \\ (x2,y2)=(2,3) \\ m=(y2-y1)/(x2-x1)=(3-0)/(2-0)=(3)/(2) \end{gathered}`

Using the point slope equation:

`\begin{gathered} y-y1=m(x-x1) \\ y-0=(3)/(2)(x-0) \\ y=(3)/(2)x \end{gathered}`

Equation 2:

Let:

`\begin{gathered} (x1,y1)=(2,3) \\ (x2,y2)=(4,6) \\ m=(y2-y1)/(x2-x1)=(6-3)/(4-2)=(3)/(2) \end{gathered}`

Using the point slope equation:

`\begin{gathered} y-y1=m(x-x1) \\ y-3=(3)/(2)(x-2) \\ y-3=(3)/(2)x-3 \\ y=(3)/(2)x \end{gathered}`

Since the equation 1 is equal to the equation 2, we can conclude that there are Infinitely Many Solutions​