Math Problem Statement

"Find two numbers whose sum is 81 and whose difference is 35."

Solution

Let the two numbers be xx and yy.

Given:

  1. x+y=81x + y = 81 (Their sum is 81)
  2. xy=35x - y = 35 (Their difference is 35)

We now have two equations:

  1. x+y=81x + y = 81
  2. xy=35x - y = 35

To solve this system of equations, we can add the two equations together:

(x+y)+(xy)=81+35(x + y) + (x - y) = 81 + 35

Simplifying:

2x=1162x = 116

Solving for xx:

x=1162=58x = \frac{116}{2} = 58

Now substitute x=58x = 58 into the first equation x+y=81x + y = 81:

58+y=8158 + y = 81

Solving for yy:

y=8158=23y = 81 - 58 = 23

Thus, the two numbers are 5858 and 2323.

Would you like more details on this solution or have any other questions?

Here are 5 related questions to consider:

  1. What is the sum and difference of two numbers when given other constraints?
  2. How can you use substitution to solve systems of equations?
  3. What methods can you use to solve simultaneous equations?
  4. Can you derive other relationships between numbers if their sum and difference are known?
  5. How do different algebraic techniques simplify solving pairs of equations?

Tip: When solving systems of equations, always look for simple operations like addition or subtraction that eliminate one variable.

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Math Problem Analysis

Mathematical Concepts

Algebra
Systems of Linear Equations

Formulas

x + y = sum
x - y = difference
2x = sum + difference

Theorems

Solving systems of linear equations by addition

Suitable Grade Level

Grades 6-8