One equation already gives $y$ in terms of $x$. Try substituting $y = 2x - 1$ into the other equation so you have just one variable.

After substitution, you should get a quadratic equation in $x$. Think about how the number of real solutions to a quadratic relates to its discriminant $b^2 - 4ac$.

Each real solution for $x$ will produce exactly one $y$-value from the equation $y = 2x - 1$. Once you know how many real $x$-values there are, you can tell how many ordered pairs $(x, y)$ satisfy the system.

In Desmos, type x^2 + y^2 = 13 to graph the circle centered at the origin with radius sqrt(13).

In a new line, type y = 2x - 1 to graph the straight line.

Look for where the line and the circle cross. Click on each intersection point Desmos highlights to see its coordinates, and count how many distinct intersection points there are. That count is the number of ordered pairs solving the system.

The first equation, $x^2 + y^2 = 13$, represents a circle centered at the origin with radius $\sqrt{13}$. The second equation, $y = 2x - 1$, is a straight line. Algebraically, we can find their intersection points by substitution; geometrically, we are finding how many points the line and the circle share.

Since $y = 2x - 1$, substitute this expression for $y$ into $x^2 + y^2 = 13$:

Now expand and simplify:

Combine like terms:

Move 13 to the left side:

So we have a quadratic equation in $x$.

For the quadratic $5x^2 - 4x - 12 = 0$, the coefficients are $a = 5$, $b = -4$, and $c = -12$.

Compute the discriminant $\Delta = b^2 - 4ac$:

We will use this value to find the solutions for $x$ in the next step.

Use the quadratic formula $x = \dfrac{-b \pm \sqrt{\Delta}}{2a}$ with $a = 5$, $b = -4$, and $\Delta = 256$:

So the two values of $x$ are

Now find the corresponding $y$-values using $y = 2x - 1$:

• If $x = 2$, then $y = 2(2) - 1 = 3$.
• If $x = -\dfrac{6}{5}$, then

So the system has the two ordered pairs $(2, 3)$ and $\left(-\dfrac{6}{5}, -\dfrac{17}{5}\right)$. Therefore, the number of ordered pairs that satisfy the system is exactly two.