Find parametric equations for the line of slope 3 that passe

Find parametric equations for the line of slope 3 that passes through the point (5, 7). (Use t as your parametrized variable. Enter your answers as a comma-separated list.)

Solution

This is a good problem to give students when learning about parametric equations.  The first thought a student might have is to go ahead and find the equation of the line using the point-slope form.  We need to recall that the point-slope form looks like the following:

y – y₁ = m(x – x₁)

where m = slope and (x₁, y₁) is the given point

Therefore we can substitute in the given values for the certain variables and solve in the following way.

y – 7 = 3(x – 5)

y – 7 = 3x – 15

y = 3x – 15 + 7

y = 3x – 8

So we now have our equation of the line through the point (5, 7) and with a slope of 3 to be y = 3x – 8.

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). The two equations are usually called the parametric equations of a curve. The extent of the curve will depend on the range of t and your work with parametric equations should pay close attention the range of t. In many applications, we think of x and y \"varying with time t\" or the angle of rotation that some line makes from an initial location.  In this case, we are not dealing with curves; instead we are working with a single equation for a line.  In order to write the equation of the line in parametric form, we still must have two equations and in t.  When looking at the equation y = 3x – 8, we see that we have an equation for y, but it is not in terms of t.  So how can we change it so we can use it in a parametric equation?  Well, we can substitute t in for x.  This would then give us our equation for y in terms of t; y = 3t – 8.  Now we need an equation for x in terms of t.  Due to the fact that we substituted t in for x, we know that x must equal t.  Therefore, our equation for x will be x = t.  Hence are parametric equations for a line that goes through the point (5, 7) and that has a slope of 3 is:

x = t

y = 3t – 8