Positive real numbers and satisfy and . What is the value of ?

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SAT Nonlinear Equations & Systems Question 213 | Free SAT Prep | Aniko

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Question 213·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Positive real numbers $x$ and $y$ satisfy $x+y=15$ and $x^2 - y^2 = 45$ .

What is the value of $x$ ?

For systems with expressions like $x^2 - y^2$ and a simple sum or difference like $x + y = \text{constant}$ , immediately use the identity $x^2 - y^2 = (x - y)(x + y)$ . Substitute the known value of $x + y$ to quickly find $x - y$ , then solve the resulting two linear equations ( $x + y$ and $x - y$ ) by adding or subtracting them. This avoids messy algebra and gets you to the answer efficiently.

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Use a special factoring pattern

Look at $x^2 - y^2$ . Can you rewrite this expression as a product using the difference of squares formula?

Connect the two equations

Once you write $x^2 - y^2$ as $(x - y)(x + y)$ , how can you use the fact that $x + y = 15$ to simplify the equation?

Solve a simple system

After substituting, you should get values for both $x + y$ and $x - y$ . How can you combine these two equations to solve for $x$ ?

Desmos Guide

Represent y in terms of x

In Desmos, enter the line y = 15 - x to represent the equation $x + y = 15$ .

Graph the second equation using substitution

Use the expression from step 1 in the second equation: enter f(x) = x^2 - (15 - x)^2. This is $x^2 - y^2$ written in terms of $x$ only.

Set the expression equal to 45 and find the intersection

Enter the horizontal line y = 45. Then look for the point(s) where the graph of $y = f(x)$ intersects the line y = 45. The $x$ -coordinate of the intersection that satisfies the given conditions is the value of $x$ .

Step-by-step Explanation

Rewrite the second equation using a factoring identity

The second equation is $x^2 - y^2 = 45$ . Recognize this as a difference of squares:

x^2 - y^2 = (x - y)(x + y).

So we can rewrite the equation as

(x - y)(x + y) = 45.

Use the first equation to substitute into the factored form

We are told that $x + y = 15$ . Substitute this into the factored equation:

(x - y)(15) = 45.

Now solve for $x - y$ by dividing both sides by $15$ :

x - y = \frac{45}{15}.

Find the two linear equations for x and y

From the previous step,

x - y = 3.

We also know from the problem that

x + y = 15.

So we now have a system of two linear equations:

$x + y = 15$
$x - y = 3$

Solve the system to find x

Add the two equations:

(x + y) + (x - y) = 15 + 3 \\[0.7em] 2x = 18.

Divide both sides by $2$ :

x = 9.

So, the value of $x$ is $\boxed{9}$ .

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Step-by-step Explanation

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Desmos Guide

Step-by-step Explanation