Convert the parametric equations x = 2 + 4t/5, y = 3t/5 - 1 into a Cartesian equation. (5 points)
A) y = 3x/4
B) y = 3x/4 - 5/2
C) y = 3x/4 - 1/2
D) y = 3x/4 + 1/2
x = 2 + 4t/5, y = 3t/5 - 1
5 x = 10 + 4 t
4 t = 5 x - 10
t = 1.25 x - 2.5
and
y = 3t/5 - 1 ===> y = 3 (1.25x-2.5)/5 - 5/5
5 y = 3.75 x - 7.5 - 5 = 3.75 x - 12.5
y = .75 x - 2.5 = 3 x/4 - 5/2
To convert parametric equations into Cartesian equations, we need to eliminate the parameter (t) by expressing it in terms of x and y.
Given the parametric equations:
x = 2 + (4t/5)
y = (3t/5) - 1
Step 1: Isolate t in one of the equations.
From the first equation, we can isolate t:
x - 2 = (4t/5)
(5/4)(x - 2) = t
Step 2: Substitute the expression for t in the other equation.
Now, substitute (5/4)(x - 2) for t in the second equation:
y = (3((5/4)(x - 2))/5) - 1
Step 3: Simplify the equation.
Let's simplify the equation:
y = (15/20)(x - 2) - 1
y = (3/4)(x - 2) - 1
y = (3/4)x - 6/4 - 1
y = (3/4)x - 7/4
Therefore, the Cartesian equation is:
y = (3/4)x - 7/4
Looking at the answer choices:
A) y = 3x/4
B) y = 3x/4 - 5/2
C) y = 3x/4 - 1/2
D) y = 3x/4 + 1/2
The correct answer is option C) y = 3x/4 - 1/2