The difference of two numbers is 5. Three times the larger, decreased by 5 times the smaller is 7. Find the numbers

X = The larger n#.

Y = The smaller #.

Eq2: X - Y = 5.
Eq2: 3X - 5Y = 7
Multiply Eq1 by -3:
-3X +3Y = -15
3X - 5Y = 7
Sum: -2y = -8,
Y = 4.

In Eq1,substitute 4 for Y:
x - 4 = 5,
X = 9.

To find the numbers, let's assign variables to represent the two unknown numbers. Let's say the larger number is represented by 'x' and the smaller number is represented by 'y'.

According to the problem, "The difference of two numbers is 5," which can be expressed as an equation:

x - y = 5 -- Equation 1

The problem also states that "Three times the larger, decreased by 5 times the smaller is 7." This can be expressed as another equation:

3x - 5y = 7 -- Equation 2

Now, we have a system of two equations with two unknowns. To solve this system, we can utilize a method called substitution.

First, let's solve Equation 1 for 'x' in terms of 'y':

x = y + 5

Substitute this expression for 'x' into Equation 2:

3(y + 5) - 5y = 7

Simplify and solve for 'y':

3y + 15 - 5y = 7
-2y + 15 = 7
-2y = 7 - 15
-2y = -8
y = -8 / -2
y = 4

Now, substitute the value of 'y' back into Equation 1 to find 'x':

x - 4 = 5
x = 5 + 4
x = 9

So, the two numbers are 9 and 4.