Problem:
Two numbers differ by 5. If their sum is 19, find the two numbers.

Step 1: Understand the problem
We are given two numbers that differ by 5, and their sum is 19. We need to find the two numbers.

Step 2: Define the variables
Let's assume the two numbers as x and y. We are given that x and y differ by 5 and their sum is 19.

Step 3: Formulate the equations
We can write two equations based on the given information:
1. x - y = 5 (since x and y differ by 5)
2. x + y = 19 (since their sum is 19)

Step 4: Solve the equations
We can solve these equations simultaneously to find the values of x and y.

Method 1: Elimination method
Using the elimination method, we can add the two equations to eliminate the variable y.

(x - y) + (x + y) = 5 + 19
2x = 24
x = 12

Substituting the value of x in equation 2:
12 + y = 19
y = 7

So, the two numbers are 12 and 7.

Method 2: Substitution method
Using the substitution method, we can solve one equation for one variable and substitute it into the other equation.

From equation 1, we can express x in terms of y:
x = y + 5

Substituting x in equation 2:
(y + 5) + y = 19
2y + 5 = 19
2y = 14
y = 7

Substituting the value of y in equation 1:
x - 7 = 5
x = 12

So, the two numbers are 12 and 7.

Step 5: Check the solution
We can verify our solution by substituting the values of x and y in the given equations:
1. x - y = 5
12 - 7 = 5 (True)
2. x + y = 19
12 + 7 = 19 (True)

Therefore, the solution is correct.

Conclusion:
The two numbers are 12 and 7. The solution was obtained by solving the given equations using either the elimination or substitution method. The solution was then verified by substituting the values in the equations.

Difference=5
sum=19
let the first number=x
let the second number=19-x
ATQ,
19-x-x=5
19-2x=5
-2x=5-19
-2x=-14
x=14/2
x=7
so the first number=x=7
second number=19-x=19-7=12.