To solve this problem, we'll use a system of linear equations. We are given two pieces of information:
1. The sum of two numbers is 95.
2. One number exceeds the other by 13.
Let's denote the two numbers as [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. According to the given conditions, we can write two equations:
1. [tex]\( x + y = 95 \)[/tex]
2. [tex]\( x - y = 13 \)[/tex]
### Step-by-Step Solution:
1. Write down the two equations:
[tex]\[
\begin{cases}
x + y = 95 \quad \text{(Equation 1)}\\
x - y = 13 \quad \text{(Equation 2)}
\end{cases}
\][/tex]
2. Add the two equations together to eliminate [tex]\( y \)[/tex]:
Adding Equation 1 and Equation 2:
[tex]\[
(x + y) + (x - y) = 95 + 13
\][/tex]
This simplifies to:
[tex]\[
x + x = 108
\][/tex]
Which means:
[tex]\[
2x = 108
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
Dividing both sides by 2:
[tex]\[
x = \frac{108}{2}
\][/tex]
So:
[tex]\[
x = 54
\][/tex]
4. Substitute [tex]\( x = 54 \)[/tex] back into Equation 1 to solve for [tex]\( y \)[/tex]:
Using [tex]\( x + y = 95 \)[/tex]:
[tex]\[
54 + y = 95
\][/tex]
Subtract 54 from both sides:
[tex]\[
y = 95 - 54
\][/tex]
So:
[tex]\[
y = 41
\][/tex]
### Conclusion:
The two numbers are [tex]\( 54 \)[/tex] and [tex]\( 41 \)[/tex]. They satisfy the conditions that their sum is 95 and one exceeds the other by 13. Therefore, the numbers are [tex]\( 54 \)[/tex] and [tex]\( 41 \)[/tex].
