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:

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

Using the point slope equation:

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

Equation 2:

Let:

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

Using the point slope equation:

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

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