The problem:

Write parametric equations of a line segment through (7, 5) with slope of 3.  Graph the line segment using your equations.

Recall the point-slope form, an equation for the line is

y - y1 = m ( x - x1).

If we let x = t, the y - y1 = m (t - x1) and we obtain the parametrization

x = t, y - y1 = m (t - x1 ) ; t in R.

We can obtain another parametrization for the line if we let x - x1 = t in R.

In this case y - y1 = mt, and we have

x = x1 + t, y = y1 + mt; t in R

In order to begin this problem.  In general, a pair of parametric equations is a pair of continuous functions that define the x- and y- coordinates of a point in a coordinate plane in terms of a third variable, such as t, called the parameter.

Thus, a parametric curve in the plane is a pair of functions

x = f (t)

y = g (t)

where the two continuous functions define ordered pairs (x, y).  Substituting into : x = x1 + t, y = y1 + mt; t in R we get:

When I put these back in point-slope form I obtain the following:

  y - 5 = 3 (x - 7)

y = 3x - 21 + 5

                                                    y = 3x - 16

So, both the parametric equations and y = 3x - 16 should graph the same line.  See below.


As you can see above these are the same lines.


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