Consider the system of equations: What is one possible value of ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 109 | Free SAT Prep | Aniko

00:00

Question 109·Easy·Nonlinear Equations in One Variable; Systems in Two Variables

0 / 233

Consider the system of equations:

\begin{aligned} x^2 + y^2 &= 25 \\ y &= 3 \end{aligned}

What is one possible value of $x$ ?

For systems where one equation is nonlinear (like a circle) and the other is a simple linear equation giving $x$ or $y$ directly, use substitution: plug the known value into the nonlinear equation to get a single-variable equation. Solve carefully, and when you get something like $x^2 = k$ , always remember to consider both the positive and negative square roots unless the context clearly restricts the variable. This approach is fast, avoids unnecessary graphing, and works reliably on SAT grid-in questions.

Hints

Click me!

Use the equation that is already solved for a variable

One of the equations already tells you the value of $y$ . How can you use $y = 3$ in the other equation?

Rewrite the first equation using $y = 3$

After substituting $y = 3$ into $x^2 + y^2 = 25$ , you should get an equation involving only $x$ . What does that equation look like?

Solve the resulting equation for $x$

Once you have $x^2 + 9 = 25$ , isolate $x^2$ by moving $9$ to the other side. Then think: which numbers, when squared, give that value? Remember there may be two solutions.

Desmos Guide

Graph the circle

Type x^2 + y^2 = 25 into Desmos. This graphs a circle of radius 5 centered at the origin.

Graph the horizontal line

On a new line, type y = 3. This will draw a horizontal line crossing the circle.

Find the intersection points

Look at the points where the line $y = 3$ intersects the circle. Read off the $x$ -coordinate(s) of these intersection points; these are the possible values of $x$ that satisfy the system.

Step-by-step Explanation

Substitute using the second equation

You are given the system:

x^2 + y^2 = 25 \\[0.7em] y = 3

Use the second equation to replace $y$ with $3$ in the first equation:

x^2 + 3^2 = 25.

Simplify and isolate $x^2$

First simplify $3^2$ and then solve for $x^2$ :

x^2 + 3^2 = 25 \\[0.7em] x^2 + 9 = 25 \\[0.7em] x^2 = 25 - 9 \\[0.7em] x^2 = 16.

Take square roots to solve for $x$

To solve $x^2 = 16$ , take the square root of both sides. Remember that both the positive and negative roots are solutions when you solve $x^2 = a$ :

x = \pm 4.

So the two possible values of $x$ are $4$ and $-4$ . Since the question asks for one possible value of $x$ , you can enter $4$ as a correct answer.

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation