A system of equations is given. What is the positive value of that satisfies the system?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 82 | Free SAT Prep | Aniko

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Question 82·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

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A system of equations is given.

\begin{aligned} x^2 + y &= 10 \\ y &= 6 \end{aligned}

What is the positive value of $x$ that satisfies the system?

Your answer

(Express the answer as an integer)

For systems where one equation already gives a variable (like $y = 6$ ), immediately use substitution: plug that value into the other equation to get a single-variable equation. Then solve step by step, and if you get $x^2 = k$ , remember it has two solutions, $x = \pm\sqrt{k}$ , and carefully choose the one that matches the question (positive, negative, or both). This avoids unnecessary graphing or complicated algebra and saves time on the SAT.

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Start with the easy equation

Look at the second equation. Does it already tell you the value of one variable directly?

Substitute into the first equation

Once you know $y$ , replace $y$ in the first equation with that value. What equation in terms of only $x$ do you get?

Solve the resulting equation carefully

After isolating $x^2$ , remember that solving $x^2 = k$ gives two possible values for $x$ . Then decide which one fits what the question is asking for.

Desmos Guide

Enter the equations

In Desmos, type y = 6 on one line and y = 10 - x^2 on another line. These represent the same system of equations in a more graphable form.

Find the intersection points

Look for the points where the horizontal line $y = 6$ intersects the curve $y = 10 - x^2$ . Tap on the intersection points and note the positive $x$ -coordinate shown there; that is the value of $x$ that the question is asking for.

Step-by-step Explanation

Use the simpler equation

From the second equation, we are directly given the value of $y$ :

y = 6

We will use this value in the first equation.

Substitute into the first equation

Substitute $y = 6$ into the first equation $x^2 + y = 10$ :

x^2 + 6 = 10

Now solve this equation for $x$ .

Solve for $x^2$

Isolate $x^2$ by subtracting $6$ from both sides:

x^2 + 6 = 10 \\[0.7em] x^2 = 10 - 6 \\[0.7em] x^2 = 4

Now take the square root of both sides to find $x$ .

Find the positive value of x

If $x^2 = 4$ , then

x = \pm 2

The problem asks for the positive value of $x$ , so the answer is $x = 2$ .

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