The aim of the given question is to introduce **word problems** related to **basic algebra** and** arithmetic operations**.

To solve such questions we may need to **first assume** the required numbers as **algebraic variables**. Then we try to **convert the given constraints** into the form of **algebraic equations**. Finally, we **solve these equations** to find the values of the **required numbers**.

## Expert Answer

**Let** $ x $ **be the number** that we want to find. Then:

\[ \text{ Product of } x \text{ and } 7 \ = \ ( x )( 7 ) \ = \ 7 x \]

**And:**

\[ \text{ Two more than } x \ = \ x \ + \ 2 \]

Under the **given conditions and constraints**, we can formulate the following equation:

\[ \text{ Product of } x \text{ and } 7 \ = \ \text{ Two more than } x \]

\[ \Rightarrow 7 x \ = \ x \ + \ 2 \]

**Subtracting** $ x $ from both sides:

\[ 7 x \ – \ x \ = \ x \ + \ 2 \ – \ x \]

\[ \Rightarrow 6 x \ = \ 2 \]

**Dividing** both sides by $ 6 $:

\[ \dfrac{ 1 }{ 6 } \times 6 x \ = \ \dfrac{ 1 }{ 6 } \times 2 \]

\[ \Rightarrow x \ = \ \dfrac{ 1 }{ 3 } \]

Which is the required number.

## Numerical Result

\[ x \ = \ \dfrac{ 1 }{ 3 } \]

## Example

Find **two number**s such that the **sum of both numbers is equal to 2 more than their product** and **one of the numbers is 2 more than the other** number.

**Let** $ x $ and $ y $ be the **number that we want to find**. Then:

\[ \text{ Two more than product of } x \text{ and } y \ = \ ( x )( y ) \ + \ 2 \ = \ x y \]

\[ \text{ Sum of } x \text{ and } y \ = \ x \ + \ y \ = \ \]

**And:**

\[ \text{ Two more than } x \ = \ x \ + \ 2 \]

Under the **given conditions and constraints**, we can formulate the following equations:

\[ \text{ Sum of } x \text{ and } y \ = \ \text{ Two more than product of } x \text{ and } y \]

\[ x \ + \ y \ = \ x y \ + \ 2 \ … \ … \ … \ ( 1 ) \]

**And:**

\[ x \ = \ y \ + \ 2 \ … \ … \ … \ ( 2 ) \]

**Substituting** the value of $ x $ from e**quation (2) in equation (1)**:

\[ ( y \ + \ 2 ) \ + \ y \ = \ ( y \ + \ 2 ) y \ + \ 2 \]

\[ \Rightarrow 2 y \ + \ 2 \ = \ y^2 \ + \ 2 y \ + \ 2 \]

**Adding** $ – 2 y – 2 $ on both sides:

\[ 2 y \ + \ 2 \ – \ 2 y \ – \ 2 = \ y^2 \ + \ 2 y \ + \ 2 \ – \ 2 y \ – 2 \]

\[ \Rightarrow 0 \ = \ y^2 \]

\[ \Rightarrow y \ = \ 0 \]

**Substituting** this value of $ y $ **in equation (2):**

\[ x \ = \ ( 0 ) \ + \ 2 \]

\[ \Rightarrow x \ = \ 2 \]

Hence, **0 and 2 are the required numbers.**