Let's set up and solve the system of equations step-by-step:
1. Identify the Variables:
- Let [tex]\( x \)[/tex] be the first integer.
- Let [tex]\( y \)[/tex] be the second integer.
2. Set up the Equations:
- According to the problem, the sum of the two integers is 41. This gives us the first equation:
[tex]\[ x + y = 41 \][/tex]
- The problem also states that the difference between the two integers is 5. This gives us the second equation:
[tex]\[ x - y = 5 \][/tex]
3. Write the System of Equations:
[tex]\[
\begin{cases}
x + y = 41 \quad \ \ \ (1)\\
x - y = 5 \ \ \ \ \quad (2)
\end{cases}
\][/tex]
4. Solve the System of Equations:
- We will solve this system by adding and subtracting the equations to eliminate one of the variables.
- First, add the two equations (1) and (2):
[tex]\[
(x + y) + (x - y) = 41 + 5
\][/tex]
- Simplify this:
[tex]\[
x + y + x - y = 46
\][/tex]
[tex]\[
2x = 46
\][/tex]
[tex]\[
x = \frac{46}{2}
\][/tex]
[tex]\[
x = 23
\][/tex]
- Now, use the value of [tex]\( x \)[/tex] in either equation (1) or (2) to find [tex]\( y \)[/tex]. Let's use equation (1):
[tex]\[
x + y = 41
\][/tex]
[tex]\[
23 + y = 41
\][/tex]
[tex]\[
y = 41 - 23
\][/tex]
[tex]\[
y = 18
\][/tex]
5. Conclusion:
- The solution to the system of equations is [tex]\((x, y) = (23, 18)\)[/tex].
- Therefore, the two integers are [tex]\( 23 \)[/tex] and [tex]\( 18 \)[/tex].
