To find the two numbers, let's use the information provided:
We know:
1. The sum of the two numbers is 37.
2. The difference of the two numbers is 9.
Let's call the two numbers [tex]\(x\)[/tex] and [tex]\(y\)[/tex], where [tex]\(x\)[/tex] is the larger number, and [tex]\(y\)[/tex] is the smaller number.
We can write two equations based on the information given:
[tex]\[ x + y = 37 \quad \text{(Equation 1)} \][/tex]
[tex]\[ x - y = 9 \quad \text{(Equation 2)} \][/tex]
### Step-by-Step Solution:
1. Add the two equations to eliminate [tex]\(y\)[/tex]:
[tex]\[ (x + y) + (x - y) = 37 + 9 \][/tex]
[tex]\[ x + y + x - y = 46 \][/tex]
[tex]\[ 2x = 46 \][/tex]
2. Solve for [tex]\(x\)[/tex]:
[tex]\[ x = \frac{46}{2} \][/tex]
[tex]\[ x = 23 \][/tex]
So, the larger number [tex]\(x\)[/tex] is 23.
3. Substitute [tex]\(x = 23\)[/tex] back into Equation 1 to find [tex]\(y\)[/tex]:
[tex]\[ 23 + y = 37 \][/tex]
[tex]\[ y = 37 - 23 \][/tex]
[tex]\[ y = 14 \][/tex]
So, the smaller number [tex]\(y\)[/tex] is 14.
### Conclusion
The two numbers are:
- Larger number: 23
- Smaller number: 14
