SAT Nonlinear Equations & Systems Question 160 | Free SAT Prep | Aniko

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Question 160·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{aligned} y &= 2x \\ x^2 + y &= 15 \end{aligned}

If $x$ is positive and the system above is true, what is the value of $x$ ?

For systems where one equation is already solved for a variable (like $y=2x$ ), use substitution: plug that expression into the other equation to get a single equation in one variable. Rearrange to standard quadratic form, factor if possible, and solve. Finally, apply any extra conditions (such as $x$ being positive) to choose the correct solution quickly from the options.

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Relate the two equations

The first equation tells you that $y=2x$ . How can you use that to replace $y$ in the second equation?

Form a single-variable equation

After substituting for $y$ , make sure all terms are on one side so the equation equals $0$ . What type of equation do you get?

Factor and use the positivity condition

Factor the quadratic equation in $x$ . You will get two possible values of $x$ ; which one fits the requirement that $x$ is positive?

Desmos Guide

Enter both equations in Desmos

Type y = 2x as the first equation. Then rewrite the second equation as $y = 15 - x^2$ and type y = 15 - x^2 as the second equation.

Find the intersection points

Look at where the line $y = 2x$ intersects the parabola $y = 15 - x^2$ . There will be two intersection points, one with a negative $x$ -value and one with a positive $x$ -value.

Use the positive x-coordinate

Click the intersection point that lies to the right of the $y$ -axis (where $x>0$ ). The $x$ -coordinate of this point is the value of $x$ that satisfies the system and the condition that $x$ is positive.

Step-by-step Explanation

Use substitution to get an equation in one variable

You are given the system

\begin{aligned} y &= 2x \\ x^2 + y &= 15 \end{aligned}

Since $y=2x$ , substitute $2x$ for $y$ in the second equation:

x^2 + 2x = 15

Now you have an equation with only $x$ .

Rewrite in standard quadratic form and factor

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

x^2 + 2x - 15 = 0

Now factor the quadratic. You want two numbers that multiply to $-15$ and add to $2$ . Those numbers are $5$ and $-3$ , so the factored form is

(x + 5)(x - 3) = 0

Solve and use the condition that x is positive

Set each factor equal to $0$ :

$x + 5 = 0$ gives $x = -5$ .
$x - 3 = 0$ gives $x = 3$ .

The problem states that $x$ is positive, so you must choose the positive solution. Therefore, the value of $x$ is $3$ .

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