In the system of equations above, and are real numbers. What is the positive value of that satisfies the system?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 28 | Free SAT Prep | Aniko

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Question 28·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{cases} y = x^2 - 4x + 3 \\ y = x + 3 \end{cases}

In the system of equations above, $x$ and $y$ are real numbers. What is the positive value of $x$ that satisfies the system?

Your answer

(Express the answer as an integer)

For systems where a line and a quadratic are both solved for $y$ , quickly set the right-hand sides equal to form a single quadratic equation in $x$ . Move all terms to one side, simplify carefully, and factor (or use the quadratic formula if necessary). Solve for both roots, then check the question’s wording to see whether it wants all solutions, only real solutions, or a specific one (such as the positive root or larger root) so you report the correct value without doing extra work.

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Connect the two equations

Since both expressions equal $y$ , what equation can you write that relates $x^2 - 4x + 3$ and $x + 3$ ?

Create a single quadratic equation

After you set the expressions equal, move all the terms to one side so that $0$ is on the other side. What quadratic equation in $x$ do you get?

Factor to find possible x-values

Your quadratic should have a common factor. Factor it, then use the fact that if a product is $0$ , at least one of the factors must be $0$ .

Answer what is asked

You will find two $x$ -values that satisfy the system. The question only wants the positive one; which one is that?

Desmos Guide

Enter the equations

Type y = x^2 - 4x + 3 on one line and y = x + 3 on another line in Desmos so you can see both graphs.

Find the intersection points

Click (or tap) where the parabola and the line intersect; Desmos will show the coordinates of the intersection points.

Identify the requested x-value

Look at the $x$ -coordinates of the intersection points and choose the one that is positive; that is the value of $x$ that satisfies the question.

Step-by-step Explanation

Set the equations equal to each other

Because both expressions equal $y$ , any solution must make them equal to each other. Set the right-hand sides equal:

x^2 - 4x + 3 = x + 3

Move all terms to one side

Subtract $x$ and $3$ from both sides to get a quadratic equation equal to $0$ :

x^2 - 4x + 3 - x - 3 = 0

Combine like terms:

x^2 - 5x = 0

Factor the quadratic

Factor out the common factor $x$ from $x^2 - 5x$ :

x^2 - 5x = x(x - 5)

So the equation becomes

x(x - 5) = 0

Solve for x and choose the positive solution

Use the zero-product property: if $x(x - 5) = 0$ , then either

$x = 0$ , or
$x - 5 = 0$ , which gives $x = 5$ .

The question asks for the positive value of $x$ , so the answer is $5$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation