Convert the parametric equations 𝑥 is equal to two 𝑡 plus one and 𝑦 is equal to 𝑡 minus four to the rectangular form.
The question starts by giving us a pair of parametric equations and asks us to convert these into the rectangular form. This means what we need to do is take our pair of parametric equations and eliminate the variable 𝑡 so we have one single equation which is equivalent to our pair of parametric equations. There are a multiple of different ways of doing this.
One thing we could try is to make 𝑡 the subject of one of our parametric equations by rearranging. And we see that we can actually do this in this case. For example, if we add four to both sides of our equation 𝑦 is equal to 𝑡 minus four, which we can see gives us that 𝑡 is equal to 𝑦 plus four. We can then substitute the equation 𝑡 is equal to 𝑦 plus four into our parametric equation for 𝑥. This gives us that 𝑥 is equal to two multiplied by 𝑦 plus four plus one. If we then distribute the parentheses, we get that 𝑥 is equal to two 𝑦 plus two multiplied by four, which is eight plus one, which we can simplify to give us that 𝑥 is equal to two 𝑦 plus nine.
Now, we could stop here. But in the rectangular form, it’s standard to try to write 𝑦 as the subject of our equations. So let’s rearrange our equation 𝑥 is equal to two 𝑦 plus nine to make 𝑦 the subject. We’ll start by subtracting nine from both sides of our equation, which gives us that 𝑥 minus nine is equal to two 𝑦. Next, we can multiply both sides of our equation by a half.
This gives us that 𝑥 minus nine all divided by two is equal to two 𝑦 divided by two. And we can cancel the shared factors of two in the numerator and the denominator to just get 𝑦. Giving us that if we convert the parametric equations 𝑥 is equal to two 𝑡 plus one and 𝑦 is equal to 𝑡 minus four into the rectangular form, we get that 𝑦 is equal to 𝑥 minus nine all divided by two.