The Problem:

The problem states that the sum of two numbers is 80, and the greater number exceeds twice the smaller number by 11. We need to find the two numbers.

Let's assume:

Let's assume that the smaller number is 'x' and the greater number is 'y'. We can then proceed to solve the problem using these variables.

The Given Information:

1. The sum of two numbers is 80: x + y = 80
2. The greater number exceeds twice the smaller number by 11: y = 2x + 11

Method of Solving:

To find the two numbers, we can use a method called substitution. This involves substituting the value of one variable in terms of the other variable into the other equation. Let's solve the system of equations step by step.

Step 1: Solve equation (1) for x
x + y = 80
x = 80 - y

Step 2: Substitute the value of x from Step 1 into equation (2)
y = 2x + 11
y = 2(80 - y) + 11
y = 160 - 2y + 11
3y = 171
y = 57

Step 3: Substitute the value of y from Step 2 into equation (1)
x + y = 80
x + 57 = 80
x = 80 - 57
x = 23

The Solution:

The two numbers are 23 and 57, with 23 being the smaller number and 57 being the greater number.

Explanation:

By substituting the value of one variable into the other equation, we were able to solve the system of equations and find the values of the two numbers. The smaller number is 23 and the greater number is 57.

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