Solve the following by setting up the proper equations to describe the facts given and then carrying out the mathematical calculations to solve for the unknown variable(s). Let [tex]x[/tex] represent the larger number and [tex]y[/tex] represent the smaller number. Enter your answer in [tex](x, y)[/tex] format.

The sum of two numbers is 44. Their difference is 12. Determine the two unknown numbers.



Answer :

Certainly! Let's follow these steps to solve the problem involving two unknown numbers, [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

### Problem Statement
We are given:
1. The sum of two numbers is 44.
2. Their difference is 12.

We need to determine these two numbers: [tex]\( x \)[/tex] (the larger number) and [tex]\( y \)[/tex] (the smaller number).

### Step-by-Step Solution

1. Set up the equations based on the given conditions:
- The sum of the two numbers: [tex]\( x + y = 44 \)[/tex]
- The difference of the two numbers: [tex]\( x - y = 12 \)[/tex]

So, we have the system of linear equations:
[tex]\[ \begin{cases} x + y = 44 \\ x - y = 12 \end{cases} \][/tex]

2. Add the two equations to eliminate [tex]\( y \)[/tex]:
[tex]\[ (x + y) + (x - y) = 44 + 12 \][/tex]

Simplifying the left-hand side, we get:
[tex]\[ x + x + y - y = 44 + 12 \][/tex]
[tex]\[ 2x = 56 \][/tex]

3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{56}{2} = 28 \][/tex]

4. Substitute the value of [tex]\( x \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
Using the equation [tex]\( x + y = 44 \)[/tex]:
[tex]\[ 28 + y = 44 \][/tex]

Solving for [tex]\( y \)[/tex]:
[tex]\[ y = 44 - 28 \][/tex]
[tex]\[ y = 16 \][/tex]

### Conclusion
The two numbers are [tex]\( x = 28 \)[/tex] and [tex]\( y = 16 \)[/tex].

So, the solution is:
[tex]\[ (x, y) = (28, 16) \][/tex]