Question

Sketch the curve represented by the parametric equations (indicate the orientation of the curve), and write the corresponding rectangular equation by eliminating the parameter. $$ x=3 t-1, \quad y=2 t+1 $$

Short answer

The curve passes through the points calculated from the parametric equations with the orientation from left to right. The rectangular equation of the curve is \(y = (2x/3) + 5/3\).

Step-by-step solution

Plotting the parametric equations

Start by sketching a set of axes and selecting some values for the parameter \(t\). For every value of \(t\) calculate \(x\) and \(y\) coordinates using the given parametric equations \(x = 3t - 1\) and \(y = 2t + 1\). This will give some points that the curve passes through. Plot these points on the graph.

Indicate orientation

The orientation of the curve is indicated by the parameter \(t\). As \(t\) increases, the points move from left to right, which shows that the orientation of the curve is from left to right.

Write the rectangular equation

To eliminate the parameter \(t\), express it from one equation and substitute it into the other. Solve \(x\) for \(t\) in the first equation, you have \(t = (x+1)/3\). Substitute this into the second equation to get \(y = 2((x+1)/3) + 1\), which simplifies to \(y = (2x/3) + 5/3\). This is the rectangular equation for the given parametric equations.