Help me with my assignment in pre calculus about System of Non Linear Equations

Question

System of Non Linear Equations: Solve the following using either substitution or elimination method. Verify your answer by sketching its graph

$3x - 2y = 0\\ x^2 + (y - 1)^2 = 11$

$8x^2 - y^2 + 16y - 18 = 0\\ 2y^2 - 48x - 16y - 3 = 0$

Answer 1

I suppose that there are two independent problems.

For the first problem, you have $$3x - 2y = 0\tag1$$ $$x^2 + (y - 1)^2 = 11\tag 2$$

From $(1)$, eliminate $y$, that is to say $y=\frac 32x$. Plug in $(2)$ to get a quadratic equation in $x$; solve for $x$ and then, for each solution, go back to $y$.

For the second problem, you have $$8x^2 - y^2 + 16y - 18 = 0\tag 3$$ $$2y^2 - 48x - 16y - 3 = 0\tag 4$$ From $(4)$,extract $x$ and plug it in $(3)$ to get a quartic equation in $y$; solve for $y$ and then, for each solution, go back to $x$.

I am sure that you already noticed that the first ptoblem is the intersection of a straight line with a circle and that the second problem is the intersection of two conics.