The system of equations is given. What is the sum of all values of that satisfy the system?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 50 | Free SAT Prep | Aniko

00:00

Question 50·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

0 / 233

The system of equations

\begin{cases} y = x^2 - 3x + 2 \\ y = 2x - 1 \end{cases}

is given.

What is the sum of all values of $x$ that satisfy the system?

Your answer

(Express the answer as an integer)

When both equations in a system are written as y equals something, immediately set the right-hand sides equal to get a single equation in x. For a line–parabola system, this gives a quadratic; if the question asks for the sum of x-values (rather than each value), use the sum-of-roots formula $-b/a$ from the quadratic’s standard form instead of fully solving, which saves time and reduces algebra mistakes.

Hints

Click me!

Connect the two equations

Since both equations are equal to $y$ , think about how you can directly relate the two expressions involving $x$ .

Form a single equation in x

Set $x^2 - 3x + 2$ equal to $2x - 1$ and then move all terms to one side so that you get a quadratic equal to 0.

Use properties of quadratics

Once you have the quadratic equation in standard form, recall that there is a quick way to find the sum of its solutions using the coefficients, without solving for each solution individually.

Desmos Guide

Graph both equations

In Desmos, enter y = x^2 - 3x + 2 on one line and y = 2x - 1 on another. Look for the points where the parabola and the line intersect; note the x-coordinates of these intersection points.

Add the x-coordinates

After identifying the two x-values from the intersection points, type an expression in Desmos that adds them (for example, x1 + x2 using the intersection labels or by entering the two numbers you see). The result shown by Desmos is the sum of all x-values that satisfy the system.

Step-by-step Explanation

Set the equations equal to each other

Since both expressions equal $y$ , any point that satisfies the system must make the right sides equal:

x^2 - 3x + 2 = 2x - 1

This equation will give the x-values where the parabola and the line intersect.

Rearrange into standard quadratic form

Move all terms to one side so the equation equals 0:

\begin{aligned} x^2 - 3x + 2 &= 2x - 1 \\ x^2 - 3x + 2 - 2x + 1 &= 0 \\ x^2 - 5x + 3 &= 0 \end{aligned}

Now we have a quadratic equation of the form $ax^2 + bx + c = 0$ with $a = 1$ , $b = -5$ , and $c = 3$ .

Use the sum-of-roots formula

For any quadratic equation $ax^2 + bx + c = 0$ , if its solutions are $x_1$ and $x_2$ , then the sum of the solutions is

x_1 + x_2 = -\frac{b}{a}.

Here, $a = 1$ and $b = -5$ , so

x_1 + x_2 = -\frac{-5}{1} = 5.

Therefore, the sum of all values of $x$ that satisfy the system is $5$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation