Question

Eliminate the parameter and obtain the standard form of the rectangular equation. $$ \begin{aligned} &\text { Line through }\left(x_{1}, y_{1}\right) \text { and }\left(x_{2}, y_{2}\right)\\\ &x=x_{1}+t\left(x_{2}-x_{1}\right), \quad y=y_{1}+t\left(y_{2}-y_{1}\right) \end{aligned} $$

Short answer

The rectangular form of the given line is \( (x - x_{1}) (y_{2} - y_{1}) = (y - y_{1}) (x_{2} - x_{1}) \)

Step-by-step solution

Solve the equations for t

First, solve both of the given equations for parameter t. From the first equation we find that \( t = (x - x_{1}) / (x_{2} - x_{1}) \). From the second equation we find that \( t = (y - y_{1}) / (y_{2} - y_{1}) \).

Set the expressions equal to each other

Following the elimination of t, set the right-hand sides of both equations equal to each other, because both are equal to t. This gives us \( (x - x_{1}) / (x_{2} - x_{1}) = (y - y_{1}) / (y_{2} - y_{1}) \).

Rearrange equation and simplify

Now, cross-multiply and afterwards simplify the equation, to find the final solution. The cross-multiply gives us \( (x - x_{1}) (y_{2} - y_{1}) = (y - y_{1}) (x_{2} - x_{1}) \). It is the rectangular form of the given line.