Two real numbers and satisfy the system - - What is ?

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SAT Nonlinear Equations & Systems Question 176 | Free SAT Prep | Aniko

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Question 176·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Two real numbers $x$ and $y$ satisfy the system

$x + y = 14$
$xy = 45$

What is $|x - y|$ ?

For SAT questions where you know the sum and product of two numbers but are asked for their difference, avoid solving for each number separately unless factoring is obviously easy. Instead, use identities that connect sum, product, and difference: derive or recall $(x-y)^2 = (x+y)^2 - 4xy$ , plug in the given values, and take the positive square root for the absolute value. This method is fast, reduces arithmetic mistakes, and works even when the individual numbers are not integers.

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Connect sum, product, and difference

You know $x+y$ and $xy$ but are asked for $|x-y|$ . Try working with $(x-y)^2$ instead of $|x-y|$ directly so you can use algebra to connect these quantities.

Compare $(x+y)^2$ and $(x-y)^2$

Write out $(x+y)^2$ and $(x-y)^2$ in terms of $x^2$ , $xy$ , and $y^2$ . How can you combine these expressions to eliminate $x^2$ and $y^2$ ?

Use the identity and substitute

From comparing $(x+y)^2$ and $(x-y)^2$ , you should get a formula of the form $(x-y)^2 = (x+y)^2 - 4xy$ . Plug in $x+y=14$ and $xy=45$ , find $(x-y)^2$ , and then take the positive square root to get $|x-y|$ .

Desmos Guide

Compute $|x-y|$ directly from sum and product

In Desmos, type sqrt((14)^2 - 4*45) and press Enter. The numerical result shown is the value of $|x-y|$ for this system.

Step-by-step Explanation

Relate $(x-y)^2$ to the given information

We are given $x+y=14$ and $xy=45$ and asked for $|x-y|$ .

To connect these, look at $(x-y)^2$ :

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

and compare it to $(x+y)^2$ :

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

From these two equations, we can eliminate $x^2 + y^2$ to get a formula for $(x-y)^2$ in terms of $x+y$ and $xy$ only.

Use the identity involving sum and product

Subtract the expression for $(x-y)^2$ from the expression for $(x+y)^2$ :

(x+y)^2 - (x-y)^2 = (x^2 + 2xy + y^2) - (x^2 - 2xy + y^2) = 4xy.

Rearrange this to solve for $(x-y)^2$ :

(x-y)^2 = (x+y)^2 - 4xy.

Now substitute the given values $x+y=14$ and $xy=45$ into this identity.

Compute $(x-y)^2$ and then $|x-y|$

Substitute into the formula:

(x-y)^2 = (14)^2 - 4(45).

Calculate:

(14)^2 = 196, \quad 4(45) = 180,

(x-y)^2 = 196 - 180 = 16.

Because $|x-y|$ is the positive square root of $(x-y)^2$ , we have

|x-y| = \sqrt{16} = 4.

So the correct answer is $4$ .

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