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.