Try squaring $x + y = 5$ and use the identity $(x + y)^2 = x^2 + 2xy + y^2$ to bring in the $x^2 + y^2$ term from the second equation.

Once you know both $x + y$ and $xy$, think of $x$ and $y$ as the roots of a quadratic equation whose coefficients come from that sum and product.

In Desmos, enter the first equation as y = 5 - x to graph the line representing $x + y = 5$.

Enter the second equation as x^2 + y^2 = 13; Desmos will display a circle centered at the origin with radius $\sqrt{13}$.

Look at the points where the line and the circle intersect. These intersection coordinates are the solutions $(x,y)$ to the system.

Compare the coordinates of the intersection point(s) you see in Desmos with the answer choices and choose the option that matches exactly.

We know $x + y = 5$, so

On the other hand,

Since we are also given $x^2 + y^2 = 13$, we can connect these facts in the next step.

Substitute $x^2 + y^2 = 13$ into the identity:

But $(x + y)^2 = 25$, so

Subtract 13 from both sides:

Now we know both the sum $x + y = 5$ and the product $xy = 6$.

If two numbers have sum 5 and product 6, they can be seen as the roots of a quadratic equation

So $x$ and $y$ are the roots of

Solve this quadratic to find the possible values of $x$ and $y$.

Factor the quadratic:

So $t = 2$ or $t = 3$, meaning $x$ and $y$ are $2$ and $3$ in some order. Both $(2,3)$ and $(3,2)$ satisfy the original equations, but among the answer choices only $(2,3)$ appears, so the correct ordered pair is $(2,3)$.