Sure, let's solve this step-by-step.
Given:
1. The sum of two numbers is 37.
2. The greater number exceeds the smaller number by 9.
Let's denote the smaller number by [tex]\( x \)[/tex] and the greater number by [tex]\( y \)[/tex].
From the information given, we can set up the following equations:
1. [tex]\( x + y = 37 \)[/tex] (Equation 1)
2. [tex]\( y = x + 9 \)[/tex] (Equation 2)
To find [tex]\( x \)[/tex] and [tex]\( y \)[/tex], we can use substitution.
Substitute Equation 2 into Equation 1:
[tex]\[ x + (x + 9) = 37 \][/tex]
Now, combine like terms:
[tex]\[ 2x + 9 = 37 \][/tex]
Next, isolate [tex]\( x \)[/tex] by subtracting 9 from both sides of the equation:
[tex]\[ 2x = 37 - 9 \][/tex]
[tex]\[ 2x = 28 \][/tex]
Now, divide both sides by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{28}{2} \][/tex]
[tex]\[ x = 14 \][/tex]
So, the smaller number [tex]\( x \)[/tex] is 14.
Now, to find the greater number [tex]\( y \)[/tex], substitute [tex]\( x \)[/tex] back into Equation 2:
[tex]\[ y = x + 9 \][/tex]
[tex]\[ y = 14 + 9 \][/tex]
[tex]\[ y = 23 \][/tex]
Therefore, the two numbers are:
[tex]\[
\text{Smaller number } x = 14\\
\text{Greater number } y = 23
\][/tex]
