# How do you find the parametric equation for y if x = e^t for the line passing through the points (2,1) and (-2,3)?

Jan 5, 2017

Start with the slope-intercept form of the equation of a line. Use the two points to compute the slope and intercept. Substitute ${e}^{t}$ for x.

#### Explanation:

The slope-intercept form of the equation of a line is:

$y = m x + b \text{ [1]}$

Use the two points to compute the slope:

$m = \frac{3 - 1}{- 2 - 2} = \frac{2}{-} 4 = - \frac{1}{2}$

Substitute $- \frac{1}{2}$ for m into equation [1]:

$y = - \frac{1}{2} x + b \text{ [2]}$

Substitute 2 for x, 1 for y, and then solve for b:

$1 = - \frac{1}{2} \left(2\right) + b$

$b = 2$

Substitute 2 for b into equation [2]:

$y = - \frac{1}{2} x + 2 \text{ [3]}$

Substitute ${e}^{t}$ for x into equation [3]:

$y = - \frac{1}{2} {e}^{t} + 2 \text{ [4]}$

Equation [4] is the parametric equation for y.