# How to find two numbers based on sum and difference?

## Question:

How to find two numbers based on sum and difference?

## Equations: Identifying Two Numbers

A typical algebraic problem that might be often encountered is identifying two unknown numbers given their sum and their difference. In such cases, the two numbers should be labeled as unknown variables **x** and **y**. Two equations relating the sum and the difference should be written and solved. This is done using an elimination method for simplicity.

## Answer and Explanation:

Assume that the two unknown numbers are **x** and **y**. Also, let's say their sum is:

{eq}x + y = a...(1) {/eq}

Additionally, let's say that the difference of these two numbers is:

{eq}x - y = b...(2) {/eq}

Adding equations (1) and (2) will let us eliminate the **y** variable:

{eq}2x = a + b...(3) {/eq}

Equation (3) lets us identify the number **x**:

{eq}x = \dfrac{a + b}{2} {/eq}

Using equation (1), we can solve for the **y** value:

{eq}y = a - x = a - \dfrac{a + b}{2} = \dfrac{2a - a - b}{2} = \dfrac{a - b}{2} {/eq}

Therefore, if the sum of two numbers is **a** and their difference is **b**, the two numbers are found by:

{eq}\boxed{x = \dfrac{a + b}{2},\\ y = \dfrac{a - b}{2}} {/eq}

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from High School Algebra II: Help and Review

Chapter 7 / Lesson 9