# Intersection of two parametric equations

This is a super basic question I'm sure but I can't figure it out and that's so frustrating. I must, in this homework problem (yes it is homework, so please do not give away the answer but rather make suggestions or give hints), find the intersection of these two equations:

\begin{align} y=x^2\tag{1} \end{align} and the line defined parametrically by \begin{align} L_1:\left\langle 2-t,7+4t \right\rangle\tag{2} \end{align}

I don't see how though. I've tried setting the components equal to one another, i.e. a parabola in parametric form is simply \begin{align} y=t,\:x=t^2,\tag{3} \end{align} but this doesn't work and I've also substituted the components of $L_1$ into $\left(3\right)$ but this gets me nowhere as well. Since it's probably so "basic" I've Googled it but I can't find a half-decent page on the matter.

• you want to vary the $t$ until the point $(2-t, 7+4t)$ is on the parabola $y = x^2.$ – abel Feb 8 '15 at 0:29
So, I think the most effective approach would be to think as follows: I want to find a value for $t$ such that the two coordinates $x$ and $y$ given by (2) satisfy the relation expressed in (1).
So maybe parametrizing the parabola is not helpful: the parameter $t$ you introduced in (3) has actually no relation with the one given for the line. If instead you look at (1) and say "ok, I want my $y$ to be $7+4t$, and my $x$..." Any ideas?