the vertical and horizontal positions of a zip line cable are represented by the following parametric equations. Rewrite the parametric equations by elimination the the parameter.
x(t)= 3t + 4
y(t) = 5-t
a) y = -1/3x+19/3
b) y= 19 - 3x
c) y= 2x + 9
d) y= 5-x
from x(t)= 3t + 4
3t = x-4
t = (x-4)/3
in y = 5-t
= 5 - (x-4)/3
= (15 - x + 4)/3
= 19/3 - x/3 <---- one of your choices
To eliminate the parameter, we need to solve one of the equations for t and substitute it into the other equation.
Given:
x(t) = 3t + 4
y(t) = 5 - t
To eliminate the parameter t, we can solve the first equation (x(t) = 3t + 4) for t:
3t = x - 4
t = (x - 4) / 3
Now substitute this value of t into the second equation (y(t) = 5 - t):
y = 5 - [(x - 4) / 3]
Simplifying further:
y = 5 - (x/3) + (4/3)
Combining the constants:
y = -(1/3)x + (19/3)
Therefore, the answer is:
a) y = -(1/3)x + (19/3)
To eliminate the parameter, we need to express t in terms of x (or y) and substitute it into the second equation.
Given:
x(t) = 3t + 4
y(t) = 5 - t
To eliminate t, we can solve the first equation for t:
x(t) = 3t + 4
x - 4 = 3t
t = (x - 4)/3
Now we can substitute this expression for t into the second equation:
y(t) = 5 - t
y = 5 - (x - 4)/3
Multiplying through by 3 to eliminate the fraction:
3y = 15 - (x - 4)
3y = 15 - x + 4
3y = 19 - x
Re-arranging the equation, we obtain:
x + 3y = 19
Therefore, the answer is (d) y = 5 - x, which matches the result x + 3y = 19