Question

How do you convert each parametric equation into rectangular form: $x = t - 3$, $y = 2t + 4$.

Answer 1

In a parametric equation, two variables are in the form of individual equations of any third variable. For example, as given in the question x is a function of t and at the same point y is also a function of t, whereas in a rectangular equation there are only two variables that are interdependent.

Complete step by step solution:
As given in the question we have $x = t - 3$ and at the same time, we also have $y = 2t + 4$ which are in parametric equations.
Now we have to convert these equations into rectangular equations.
So at first, we have to take $x = t - 3$ and make an equation of $t$ out of it
$\Rightarrow t = x + 3$ (Adding 3 both sides of the equation)
Now, we have to put this value of $t$ in the equation $y = 2t + 4$ and evaluate,
So we will get,
$\Rightarrow y = 2(x + 3) + 4$
$\Rightarrow y = 2x + 6 + 4$
$\Rightarrow y = 2x + 10$
Hence, this is our rectangular form of the given parametric form.

So our answer is $y = 2x + 10$.

Note: We can clearly observe that rectangular equation is just a format in which the relation of two variables, that is x and y is directly represented whereas parametric equation is a type of format where two variable are in a relationship of any third variable (common for both), as in this question, the common variable is t. We can easily convert the parametric form of an equation into a rectangular form of the equation and vice versa. They represent two different graphs in the parametric equation but when converted into a rectangular equation they represent a single graph which is different from both the earlier graphs.