Eliam C. answered • 06/15/22
Competitive Chess Coach & SAT Math Tutor
Hello Elle,
To tackle this problem, we would have to first know what a parametric equation is. I am assuming that you know what it is, but I will still explain it. A parametric equation is an equation where time with variable t is used as the independent variable of x and y. For example y=2t, or x=t+2, or x=-3t, or y=t-4 would all be parametric equations.
The problem that you give is the line given by y=2x+1 and you ask for a singular parametric equation. However, that is not possible. When there is an equation given by 2 variables, you would need a pair of parametric equations to represent the line. For example, y=x can be broken down into y=t and x=t, as when you substitute the equation x=t into y=t, you return to the original equation, y=x.
Now the question is, how do we break an equation down and turn it into a parametric equation? We can do this by setting some part of the equation for t (the variable for time). In this case, I will set 2x=t. If I do so, then the equation y=2x+1 can now be rewritten when t is substituted for 2x, thus resulting in y=t+1. Now we can go back to the equation we already have, 2x=t, and solve for x. By dividing both sides by 2, we result in x=t/2.
As can be seen, this equation of y=2x+1 can be broken into 2 parametric equations: y=t+1 and x=t/2. This is just one solution of the problem. To verify that these 2 parametric equations are correct, we can first solve for t for the first equation and get 2x=t and then substitute it into y=t+1. This results in y=2x+1, which is the original equation. Therefore, we solved the problem correctly.
If you want to understand a little more than you need to know to solve the problem, you can read below:
There are actually infinite ways of solving this problem, however. For example, if I set t=4x, then divide both sides by 2, I get t/2=2x. If we take the original equation y=2x+1 and substitute t/2=2x in, the equation becomes y=t/2+1. Solving t/2=2x for x, we would get x=t/4.
The resulting parametric equations would be y=t/2+1 and x=t/4. There are different equations than the ones earlier (y=t+1 and x=t/2) but when recombined would create the same equation. There would be infinite ways of solving this, but the easiest is still setting 2x=t as I did earlier.
I hope my explanation helps! There are a lot of math terms here that could be confusing. If you have any more questions, feel free to ask. I would love to explain more to you in a class!
-Eliam Chang
Vanderbilt '26
