Substitute $y = x + 1$ into $x^2 + y^2 = 25$, then expand $(x+1)^2$ and simplify. You should get a quadratic equation in $x$.

After solving the quadratic for $x$, use $y = x + 1$ to find each corresponding $y$. Then compute $xy$ for each solution and compare them.

In Desmos, type x^2 + y^2 = 25 to graph the circle with radius 5 centered at the origin.

Type y = x + 1 to graph the line that intersects the circle.

Click on each point where the line and the circle intersect. Desmos will display their coordinates $(x,y)$.

For each intersection point, multiply its $x$-coordinate by its $y$-coordinate (you can type an expression like x_value * y_value in a new line). Notice that both intersection points give the same product; that common value is what the question is asking for.

You are given the system

Since $y = x + 1$, substitute $x+1$ for $y$ in the first equation:

Now you have an equation involving only $x$.

Expand $(x+1)^2$ and combine like terms:

Subtract 25 from both sides:

Divide everything by 2:

Now solve this quadratic for $x$.

Factor the quadratic:

So the possible $x$-values are $x = -4$ and $x = 3$.

Use $y = x + 1$ to find each corresponding $y$:

• If $x = -4$, then $y = -4 + 1 = -3$.
• If $x = 3$, then $y = 3 + 1 = 4$.

So the two solutions to the system are $(-4, -3)$ and $(3, 4)$.

Now compute $xy$ for each solution:

• For $(-4, -3)$: $xy = (-4)(-3) = 12$.
• For $(3, 4)$: $xy = 3 \,\cdot 4 = 12$.

In both cases, $xy$ is the same. Therefore, the value that $xy$ must have for any solution of the system is 12.