Let's solve the problem step-by-step.
Given:
1. The sum of the tens digit [tex]\( x \)[/tex] and the ones digit [tex]\( y \)[/tex] is 9.
2. If the digits are reversed, the number formed is 9 less than 2 times the original number.
We can formulate the problem with the following equations:
1. [tex]\( x + y = 9 \)[/tex]
2. The original number is [tex]\( 10x + y \)[/tex]
3. The number with reversed digits is [tex]\( 10y + x \)[/tex]
4. The number with reversed digits is 9 less than 2 times the original number:
[tex]\[ 10y + x = 2(10x + y) - 9 \][/tex]
Let's solve the system of equations:
### Step 1: Solve for y in terms of x using the first equation:
[tex]\[ x + y = 9 \][/tex]
[tex]\[ y = 9 - x \][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] into the second equation:
[tex]\[ 10y + x = 2(10x + y) - 9 \][/tex]
[tex]\[ 10(9 - x) + x = 2(10x + (9 - x)) - 9 \][/tex]
### Step 3: Simplify the equation:
[tex]\[ 90 - 10x + x = 20x + 18 - 2x - 9 \][/tex]
[tex]\[ 90 - 9x = 18x + 9 \][/tex]
[tex]\[ 90 - 9 = 18x + 9x \][/tex]
[tex]\[ 81 = 27x \][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{81}{27} \][/tex]
[tex]\[ x = 3 \][/tex]
### Step 5: Substitute [tex]\( x \)[/tex] back into [tex]\( y = 9 - x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[ y = 9 - 3 \][/tex]
[tex]\[ y = 6 \][/tex]
### Step 6: Construct the original number:
The original number is:
[tex]\[ 10x + y \][/tex]
[tex]\[ 10(3) + 6 \][/tex]
[tex]\[ 30 + 6 \][/tex]
[tex]\[ 36 \][/tex]
Hence, the original number is:
[tex]\[ 36 \][/tex]
This solves the problem according to the given conditions.
