Chapter 4.1, Problem 57HP

To determine

Tocheck:If $3+\left(x·y\right)=\left(3+x\right)\left(3+y\right)$

Answer to Problem 57HP

Not a true statement.

Explanation of Solution

Given information:

  $3+\left(x·y\right)=\left(3+x\right)\left(3+y\right)$

Concept used:

Distributive Property: The product sum or difference of two or more numbers inside a parentheses with another number outside the parentheses is equivalent to the sum or difference of product of number outside the parentheses with each number inside the parenthesis. i.e.-

  $a\left(b+c-d\right)=ab+ac-ad$

Calculation:

  $\begin{array}{l}3+\left(x·y\right)=\left(3+x\right)\left(3+y\right)\\ \Rightarrow 3+xy=\left(3+x\right)3+\left(3+x\right)y\\ \Rightarrow 3+xy=9+3x+3y+xy\\ \Rightarrow 3=9+3x+3y\end{array}$

Taking $x=1,y=1$

  $\begin{array}{l}3=9+3\left(1\right)+3\left(1\right)\\ 3=15(\text{not true)}\end{array}$

Therefore $3+\left(x·y\right)\ne \left(3+x\right)\left(3+y\right)$

This was an incorrect example of distributive property, basis the definition the correct example should be-

  $3\left(x+y\right)=3x+3y$