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]
