Solve by setting up the proper equation to describe the facts given and then carrying out the mathematical calculations to solve for the unknown variable(s).

Three times a larger number added to 4 times a smaller number totals 24. When twice the smaller number is subtracted from twice the larger number, the result is 2. What are the two numbers?

x=small, y=large

3y + 4x = 24
2y - 2x = 2

x=3
y=4

Let's assume the larger number is represented by 'x' and the smaller number is represented by 'y'.

According to the given information, we can set up the following equations:

1) Three times the larger number added to four times the smaller number equals 24:
3x + 4y = 24

2) Twice the larger number minus twice the smaller number equals 2:
2x - 2y = 2

Now, we can solve these equations using either substitution or elimination method.

Let's use the elimination method to solve this system of equations.

Multiply the second equation by 2 to make the coefficients of x the same in both equations:
2(2x - 2y) = 2(2)
4x - 4y = 4

Now, we can eliminate the y term by adding the second equation and the modified second equation:
3x + 4y + 4x - 4y = 24 + 4
7x = 28

Divide both sides of the equation by 7:
x = 4

Substituting this value of x into either of the original equations, let's use the first equation:
3(4) + 4y = 24
12 + 4y = 24
4y = 24 - 12
4y = 12
y = 3

Therefore, the larger number (x) is 4 and the smaller number (y) is 3.

To solve for the two numbers, we need to set up a system of equations based on the given facts.

Let's assume that the larger number is represented by 'x' and the smaller number is represented by 'y'.

According to the first fact, "Three times a larger number added to 4 times a smaller number totals 24", we can write the equation:
3x + 4y = 24

According to the second fact, "When twice the smaller number is subtracted from twice the larger number, the result is 2", we can write the equation:
2x - 2y = 2

Now we have a system of two equations:

Equation 1: 3x + 4y = 24
Equation 2: 2x - 2y = 2

To solve this system of equations, we can use the method of substitution or elimination.

Let's use the method of elimination:
Multiply Equation 1 by 2 and Equation 2 by 3 to eliminate the 'y' terms:

Equation 1: 6x + 8y = 48
Equation 2: 6x - 6y = 6

Now, subtract Equation 2 from Equation 1:

(6x + 8y) - (6x - 6y) = 48 - 6
6x + 8y - 6x + 6y = 42
14y = 42
y = 42/14
y = 3

Substitute the value of 'y' back into Equation 1 to solve for 'x':

3x + 4(3) = 24
3x + 12 = 24
3x = 24 - 12
3x = 12
x = 12/3
x = 4

Therefore, the larger number is 4 and the smaller number is 3.