Consider the line passing through the points (2, 1) and (-2, 3). Find the parametric equation for y if x = t + 1.

See:

http://www.jiskha.com/display.cgi?id=1304142855

To find the parametric equation for y, we need to determine the equation of the line passing through the points (2, 1) and (-2, 3).

Step 1: Find the slope of the line.
The slope (m) of a line passing through two points (x₁, y₁) and (x₂, y₂) can be calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)

Given the points (2, 1) and (-2, 3), we can substitute the values into the formula:
m = (3 - 1) / (-2 - 2)
m = 2 / -4
m = -1/2

Step 2: Write the equation of the line using the slope-intercept form.
The equation of a line using the slope-intercept form is:
y = mx + b

Substituting the calculated slope (-1/2) and one of the given points (2, 1) into the equation, we can solve for the y-intercept (b):
1 = (-1/2) * 2 + b
b = 1 + 1
b = 2

Therefore, the equation of the line is:
y = (-1/2)x + 2

Step 3: Express x in terms of t.
Given that x = t + 1, we can substitute this expression into the equation of the line:
y = (-1/2)(t + 1) + 2

Simplifying the equation:
y = (-1/2)t - 1/2 + 2
y = (-1/2)t + 3/2

Therefore, the parametric equation for y is y = (-1/2)t + 3/2.

To find the parametric equation for y, we first need to find the slope of the line passing through the two given points.

The slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

m = (y2 - y1) / (x2 - x1)

Using the points (2,1) and (-2,3), the formula becomes:

m = (3 - 1) / (-2 - 2)
= 2 / -4
= -1/2

So, the slope of the line is -1/2.

Now, since the x-coordinate of the line is given as x = t + 1, we can substitute it into the equation of the line:

y - y1 = m(x - x1)

Using the point (2, 1) as (x1, y1), the equation becomes:

y - 1 = (-1/2)(t + 1 - 2)

Simplifying further:

y - 1 = (-1/2)(t - 1)

Multiplying both sides by -2 to get rid of the fraction:

-2(y - 1) = t - 1

Expanding:

-2y + 2 = t - 1

Rearranging the equation to isolate y:

-2y = t - 1 - 2
-2y = t - 3
y = (1/2)t - 3/2

Thus the parametric equation for y is y = (1/2)t - 3/2.