Answer:
(0, 1).
Method 1 (Substitution):
Substituting our two y's, we get the following:
[tex]x^2 + 5x + 1 = x^2 + 2x + 1 \Rightarrow 3x = 0 \Rightarrow x = 0[/tex]
Thus, the only set of solutions is (0, 1). A quick sketch (either by hand or on Desmos) can confirm this.
Method 2 (Elimination):
We have two equations. We'll let the top one be equation 1 and the bottom one be equation 2. Eliminating as many variables as we can, we subtract (2) from (1) to get:
0 = 3x => x = 0.
So the only set of solutions is (0, 1).
Method 3 (Gaussian elimination):
We can place this in an augmented matrix and row reduce.
[tex]\left[\begin{array}{cccc}1&5&1 & 1\\1&2&1 & 1\end{array}\right][/tex]
Row reducing this gives us:
[tex]\left[\begin{array}{cccc}1&5&1 & 1\\0&3&0 & 0\end{array}\right][/tex]
This tells us that the only solution for x is x = 0 (since we read this as "3x = 0") and thus, the only solution we get is (0, 1).
