After substitution, expand $(x+1)^2$, combine like terms, and then see if you can simplify the equation (for example, by dividing all terms by the same number).

Once you have a quadratic equation in $x$, factor it to find the two $x$-values. Then plug each $x$ back into $y = x + 1$ to get the corresponding $y$-values.

The problem does not ask for the individual solutions $(x, y)$, but for the sum of the $y$-coordinates of the solutions. After finding both $y$-values, add them together.

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

In a new line, type y = x + 1 to graph the line that will intersect the circle at two points.

Click on each point where the line and circle intersect. Desmos will display the coordinates of both intersection points; note the two $y$-values shown.

Add the two $y$-values from the intersection points (you can even enter them in Desmos as y1 + y2 using their numerical values) and use that sum as your final answer.

From the second equation, $y = x + 1$. Substitute this expression for $y$ into the circle equation $x^2 + y^2 = 25$:

Now simplify this equation in terms of $x$ only.

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

Divide every term by $2$ to make the equation simpler:

Now factor this quadratic.

Factor $x^2 + x - 12$:

So the possible $x$-values are $x = -4$ and $x = 3$. Next, find the corresponding $y$-values using $y = x + 1$.

Use $y = x + 1$ for each $x$ value:

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

So the two solutions are $(-4, -3)$ and $(3, 4)$. The question asks for the sum of the $y$-coordinates:

Therefore, the correct answer is $1$.