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(1) There is one positive (greater than 0) 2-digit integer. The sum of the tens and ones digits is 9. If the digits are reversed, the number formed is 9 less than 2 times the original number. Find the original number.

[Sol] Let [tex]$x$[/tex] be the tens digit and [tex]$y$[/tex] be the ones digit. Then, the original number is [tex]$10x + y$[/tex] and the number with reversed digits is [tex][tex]$10y + x$[/tex][/tex].

[tex]\[
\left\{
\begin{array}{l}
x + y = 9 \\
10y + x = 2(10x + y) - 9
\end{array}
\right.
\][/tex]

Note:
[tex]$36 = 10 \times 3 + 6$[/tex]
[tex]$63 = 10 \times 6 + 3$[/tex]



Answer :

GinnyAnswer
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.

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