Find the equation of a line in slope-intercept form whose parametric equations are x = 2t + 3 and y = 4t - 1
We've x = 2t+3 and y = 4t-1. Eliminate t between these two equations to obtain
y = 4(x-3)/2 -1 or y = 2x -7
To find the equation of a line in slope-intercept form, we need to determine the slope (m) and the y-intercept (b).
Given the parametric equations x = 2t + 3 and y = 4t - 1, let's solve for t in terms of y:
y = 4t - 1
Rearranging the equation:
4t = y + 1
Dividing both sides by 4:
t = (y + 1)/4
Next, we'll substitute this value of t into the equation x = 2t + 3 to get x in terms of y:
x = 2t + 3
x = 2[(y + 1)/4] + 3
x = (y + 1)/2 + 3
x = (y + 1)/2 + 6/2
x = (y + 1 + 6)/2
x = (y + 7)/2
Now that we have x and y in terms of each other, we can write the equation of the line in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.
Comparing the equation we found, y = (y + 7)/2, with the slope-intercept form, we can determine that the slope (m) is equal to 1/2.
To find the y-intercept (b), we can plug in the values of x and y into one of the parametric equations. Let's use x = 2t + 3:
When the line intersects the y-axis, x = 0. Plugging this into our equation:
0 = 2t + 3
Solving for t:
2t = -3
t = -3/2
Substituting this value of t into the equation y = 4t - 1:
y = 4(-3/2) - 1
y = -6 - 1
y = -7
Hence, the y-intercept (b) is -7.
The equation of the line in slope-intercept form is:
y = (1/2)x - 7