I don't quite understand Cartesian equations. I am told to
Eliminate the parameter to find a cartesian equation of the curve.
X= 3 - 4t, y= 2 - 3t there is no inequality
I don't understand what I am trying to find.
Actually, you are most familiar with Cartesian equations. That's just good old
y = f(x)
To convert parametric equations to Cartesian, eliminate the parameter t.
x = 3 - 4t
t = (3-x)/4
y = 2 - 3t
y = 2 - 3(3-x)/4
y = 2 - 9/4 + 3x/4
y = (3x-1)/4
Ta da.
Thank you!
To understand the concept of eliminating the parameter in the given parametric equations (X = 3 - 4t, y = 2 - 3t) and finding a Cartesian equation, let's break it down step by step.
When we have a set of parametric equations like the ones given, they represent a relation between two variables (X and y) in terms of another variable (t) known as the parameter. By eliminating the parameter, we express X and y as a function of each other without the need for the parameter anymore.
In simpler terms, eliminating the parameter in this context means finding a single equation that directly relates X and y.
Here's how you can eliminate the parameter and find the Cartesian equation:
Step 1: Start with the given parametric equations:
X = 3 - 4t
y = 2 - 3t
Step 2: Solve for t in one of the equations. Let's take the equation X = 3 - 4t:
Simplifying this equation, we get: 4t = 3 - X
Divide both sides by 4: t = (3 - X) / 4
Step 3: Substitute the value of t obtained in Step 2 into the other equation (y = 2 - 3t):
Replace t with (3 - X) / 4 in the equation y = 2 - 3t:
y = 2 - 3((3 - X) / 4)
Step 4: Simplify the equation obtained in Step 3:
Expand and simplify the equation, y = 2 - 3((3 - X) / 4):
y = 2 - (9 - 3X) / 4
y = 8/4 - (9 - 3X) / 4
y = (8 - (9 - 3X)) / 4
y = (3X - 1) / 4
Step 5: Rewrite the equation in standard Cartesian form:
Multiply both sides of the equation by 4 to eliminate the fraction:
4y = 3X - 1
Rearrange the equation to get the Cartesian equation in standard form:
3X - 4y = 1
Therefore, the Cartesian equation for the given parametric equations is 3X - 4y = 1.
By eliminating the parameter, we have found a relation between X and y without the need for the parameter t.