Question

1. when you reverse the digits in a certain two - digit number you decrease its value by 9. what is the number if the sum of its digits is 15?

Explanation:

Step1: Define variables

Let the tens digit be \( x \) and the units digit be \( y \). The two - digit number can be written as \( 10x + y \), and the number with reversed digits is \( 10y+x \).

Step2: Set up equations

We know two things:

1. When we reverse the digits, the value decreases by 9. So, \( (10x + y)-(10y + x)=9 \). Simplify this equation:

\[

$$\begin{align*} 10x + y-10y - x&=9\\ 9x-9y&=9\\ x - y&=1 \end{align*}$$

\]

1. The sum of its digits is 15. So, \( x + y=15 \).

Step3: Solve the system of equations

We have the system of equations:
\[

$$\begin{cases} x - y=1\\ x + y=15 \end{cases}$$

\]
Add the two equations together:
\[

$$\begin{align*} (x - y)+(x + y)&=1 + 15\\ 2x&=16\\ x&=8 \end{align*}$$

\]
Substitute \( x = 8 \) into the equation \( x + y=15 \):
\( 8 + y=15 \), so \( y = 15 - 8=7 \).

Step4: Find the number

The original two - digit number is \( 10x + y \). Substitute \( x = 8 \) and \( y = 7 \), we get \( 10\times8+7 = 87 \).

Answer:

The number is 87.