For real numbers and , the system of equations is satisfied. What is the value of ?

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SAT Nonlinear Equations & Systems Question 158 | Free SAT Prep | Aniko

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Question 158·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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For real numbers $x$ and $y$ , the system of equations

x + y = 5\qquad\text{and}\qquad x^2 + y^2 = 13

is satisfied. What is the value of $xy$ ?

When a system gives you $x + y$ and $x^2 + y^2$ and asks for $xy$ , avoid solving for $x$ and $y$ separately. Instead, immediately think of the identity $(x + y)^2 = x^2 + 2xy + y^2$ , plug in the given values to create a simple equation in $xy$ , and solve it. This approach is much faster and less error-prone than substitution or elimination on the original system.

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Notice the structure of the equations

You are given $x + y$ and $x^2 + y^2$ . Think about a standard algebra identity that involves both of these expressions and $xy$ .

Use the square of a sum

Recall that $(x + y)^2$ expands to $x^2 + 2xy + y^2$ . How can you use this to connect the two given equations?

Substitute and isolate $xy$

Compute $(x + y)^2$ using the given value of $x + y$ , then replace $x^2 + y^2$ with 13. After you write the equation $x^2 + 2xy + y^2 = (x + y)^2$ with numbers, solve the resulting equation for $xy$ .

Desmos Guide

Express $xy$ in terms of the given numbers

From $(x + y)^2 = x^2 + 2xy + y^2$ and the given values, you can rewrite $xy$ as

xy = \frac{(x + y)^2 - (x^2 + y^2)}{2} = \frac{25 - 13}{2}.

You will now evaluate this expression in Desmos.

Compute the value in Desmos

In Desmos, type (25 - 13)/2 on a new line. The numerical result that Desmos outputs is the value of $xy$ and is the correct answer choice.

Step-by-step Explanation

Relate the two equations using an identity

We are given $x + y = 5$ and $x^2 + y^2 = 13$ and asked for $xy$ .

Use the algebraic identity:

(x + y)^2 = x^2 + 2xy + y^2

This connects $x + y$ , $x^2 + y^2$ , and $xy$ in one equation.

Substitute the known values into the identity

From the problem, $x + y = 5$ , so

(x + y)^2 = 5^2 = 25.

Also, $x^2 + y^2 = 13$ .

Plug these into the identity $x^2 + 2xy + y^2 = (x + y)^2$ :

13 + 2xy = 25.

So we have an equation involving only $xy$ :

13 + 2xy = 25.

Solve for $xy$

Solve the equation $13 + 2xy = 25$ :

2xy = 25 - 13 = 12

xy = \frac{12}{2} = 6.

Therefore, the value of $xy$ is $\boxed{6}$ .

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation