After you set the expressions equal, move all terms to one side so that you have a quadratic equation equal to $0$. Make sure you combine like terms carefully.

Once you have a quadratic equation in standard form $ax^2 + bx + c = 0$, think about how to find the sum of its solutions without necessarily solving for each one individually. Recall the relationship between $a$, $b$, $c$ and the roots.

For a quadratic $ax^2 + bx + c = 0$, the sum of its two roots has a simple formula involving $a$ and $b$. Use that formula on your quadratic.

In Desmos, enter the two equations as:

• y = x^2 - 4x + 1
• y = 3x - 5 This will display the parabola and the line on the same coordinate plane.

Click on the points where the line and the parabola intersect. Desmos will show the coordinates of each intersection point; note the x-coordinates of these points.

In a new Desmos expression line, type the sum of the two x-coordinates you observed (for example, something like x1 + x2 using the values you saw). The value of this expression is the sum of all possible $x$-values that satisfy the system.

Because both expressions equal $y$, set them equal to each other and rearrange to standard form:

The equation $x^2 - 7x + 6 = 0$ has two solutions (the $x$-values where the graphs intersect).

The question asks for the sum of all possible values of $x$ that satisfy the system. That means we want the sum of the roots of the quadratic $x^2 - 7x + 6 = 0$.

For any quadratic equation of the form

the sum of the roots is $-\dfrac{b}{a}$.

In our equation $x^2 - 7x + 6 = 0$:

• $a = 1$
• $b = -7$

So the sum of the roots is

(You can also check by factoring $x^2 - 7x + 6 = (x - 1)(x - 6)$, whose roots $1$ and $6$ add to $7$.)

Therefore, the sum of all possible values of $x$ that satisfy the system is 7, which corresponds to choice C.