Eliminating Parameter T: Finding Cartesian Equation Y=mx+b

Hey guys! Let's dive into a super interesting problem today: how to eliminate a parameter and find the Cartesian equation of a set of parametric equations. Specifically, we're going to tackle the equations $x(t) = 17 - t$ and $y(t) = -13 - 2t$, and our goal is to get them into the familiar form of a linear equation, $y = mx + b$. Buckle up, because this is going to be fun!

Understanding Parametric Equations

First things first, let's quickly recap what parametric equations are all about. Imagine you're tracking the movement of a little robot across a screen. Instead of directly giving its $x$ and $y$ coordinates as a function of each other, we describe them separately as functions of a third variable, which we often call the parameter, $t$. Think of $t$ as time – it tells us when the robot is at a particular location. So, $x(t)$ gives the robot's horizontal position at time $t$, and $y(t)$ gives its vertical position at the same time. Our goal is to eliminate the parameter $t$ and to figure out the direct relationship between $x$ and $y$, so we can draw the robot’s path on a graph without worrying about time.

In our case, we have $x(t) = 17 - t$ and $y(t) = -13 - 2t$. These equations tell us how the $x$ and $y$ coordinates change as $t$ varies. For every value of $t$, we get a point $(x, y)$ on the coordinate plane. If we plotted all these points, we’d see a straight line. Our mission is to find the equation of that line in the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This form is super useful because it tells us everything we need to know about the line: its steepness and where it crosses the y-axis.

Now, why is this skill important? Well, eliminating the parameter isn't just a cool math trick. It's essential in many areas of physics and engineering. For example, if you're analyzing the trajectory of a projectile, you might start with parametric equations that describe its horizontal and vertical motion as functions of time. But to understand the shape of the path, you'd want to eliminate the parameter and find the equation that relates the $x$ and $y$ coordinates directly. This gives you a clear picture of the projectile's parabolic path, making it easier to predict its range and maximum height. Similarly, in computer graphics and animation, parametric equations are used to create smooth curves and paths. Eliminating the parameter can help optimize these paths and make them easier to work with. So, mastering this technique opens doors to solving a wide range of real-world problems.

Step-by-Step Guide to Eliminating the Parameter

Alright, let’s get down to business! Here’s a step-by-step guide on how to eliminate the parameter $t$ and find the Cartesian equation for our given parametric equations. We'll break it down into manageable chunks so it's crystal clear.

Step 1: Solve one equation for $t$

The first thing we need to do is pick one of our parametric equations and solve it for $t$. It doesn’t matter which equation we choose, but it’s usually easiest to pick the one that looks simpler. In our case, both equations are pretty straightforward, but let’s go with $x(t) = 17 - t$. This equation is nice and simple, and it's easy to isolate $t$. To solve for $t$, we just need to add $t$ to both sides and subtract $x$ from both sides. This gives us:

$t = 17 - x$

See? That was easy! We’ve now got an expression for $t$ in terms of $x$. This is a crucial step because it allows us to substitute this expression into the other equation and get rid of $t$ altogether. The goal here is to eliminate the parameter, and we're well on our way to doing just that. This step is like finding a key that unlocks the connection between $x$ and $y$, allowing us to see their direct relationship without the distraction of $t$. By isolating $t$, we've created a bridge that we can use to cross over from the parametric world to the Cartesian world.

Step 2: Substitute into the other equation

Now that we have $t = 17 - x$, we can substitute this expression into the other parametric equation, which is $y(t) = -13 - 2t$. This is where the magic happens! By replacing $t$ with $(17 - x)$, we’re effectively linking $y$ directly to $x$, and $t$ will vanish from the equation. This substitution is the heart of the parameter elimination process. It’s like performing a clever algebraic trick that unravels the connection between the parametric equations and reveals the underlying Cartesian equation. Let’s do the substitution:

$y = -13 - 2(17 - x)$

Notice how $t$ is gone! We now have an equation that relates $y$ and $x$ directly. But we’re not quite done yet. We need to simplify this equation and get it into the form $y = mx + b$. Think of this step as cleaning up after our algebraic magic trick. We’ve performed the substitution, but now we need to tidy up the expression so we can clearly see the relationship between $x$ and $y$. This will involve distributing the -2, combining like terms, and rearranging the equation to match the desired form. So, let’s move on to the next step and see how we can transform this equation into something beautiful and familiar.

Step 3: Simplify and rewrite in $y = mx + b$ form

Okay, we’ve got $y = -13 - 2(17 - x)$. Now it’s time to simplify this equation and get it into the form $y = mx + b$. First, let’s distribute the -2:

$y = -13 - 34 + 2x$

Next, combine the constant terms:

$y = -47 + 2x$

Finally, let’s rewrite the equation in the familiar $y = mx + b$ form. We just need to swap the terms around:

$y = 2x - 47$

And there you have it! We’ve successfully eliminated the parameter $t$ and found the Cartesian equation in the form $y = mx + b$. In this case, the slope $m$ is 2, and the y-intercept $b$ is -47. This equation tells us that the line has a positive slope, meaning it goes upwards as we move from left to right, and it crosses the y-axis at the point (0, -47). This final step is like putting the finishing touches on a masterpiece. We’ve taken a complex set of parametric equations and transformed them into a simple, elegant equation that reveals the underlying geometric shape. The $y = mx + b$ form is particularly powerful because it gives us immediate insights into the line’s characteristics, such as its slope and y-intercept.

Final Answer

So, to recap, we started with the parametric equations $x(t) = 17 - t$ and $y(t) = -13 - 2t$. By eliminating the parameter $t$, we found the Cartesian equation $y = 2x - 47$. This is a linear equation, which means the graph of these parametric equations is a straight line. The slope of the line is 2, and the y-intercept is -47.

That’s all there is to it! Eliminating the parameter might seem a bit tricky at first, but with a little practice, you’ll get the hang of it. Remember the key steps: solve one equation for $t$, substitute into the other equation, and then simplify. Once you've mastered this technique, you'll be able to tackle all sorts of problems involving parametric equations and Cartesian equations.

Keep practicing, and you'll become a pro at eliminating parameters in no time! You guys got this!