Let's break down the problem step by step:
1. We are given that the sum of the digits of a three-digit number is 13. This gives us the equation:
[tex]\[
h + t + u = 13
\][/tex]
2. The tens digit, [tex]\( t \)[/tex], is 1 more than the hundreds digit, [tex]\( h \)[/tex]. This gives us the relationship:
[tex]\[
t = h + 1
\][/tex]
3. The units digit, [tex]\( u \)[/tex], is 3 more than the sum of the tens and hundreds digits. This gives us the relationship:
[tex]\[
u = t + h + 3
\][/tex]
To formulate a system of equations, let's express all these relationships mathematically:
- We already have:
[tex]\[
h + t + u = 13
\][/tex]
- The second relationship can be rewritten as:
[tex]\[
t - h = 1
\][/tex]
- For the third relationship, let's rewrite it:
[tex]\[
u = t + h + 3
\][/tex]
By rearranging this, we get:
[tex]\[
u - t - h = 3
\][/tex]
So, our system of equations becomes:
[tex]\[
\begin{aligned}
h + t + u & = 13 \\
t - h & = 1 \\
u - t - h & = 3
\end{aligned}
\][/tex]
Simplifying them in a more familiar format, rearranging terms properly, we have:
[tex]\[
\begin{aligned}
h + t + u & = 13 \\
-h + t & = 1 \\
-h - t + u & = 3
\end{aligned}
\][/tex]
Now, let's compare this system with the provided options:
A.
[tex]\[
\begin{array}{r}
h + t + u = 13 \\
h - t = 1 \\
h + t - u = 3
\end{array}
\][/tex]
B.
[tex]\[
\begin{aligned}
h + t + u & = 13 \\
h + u & = 1 \\
-h - t & = 3
\end{aligned}
\][/tex]
C.
[tex]\[
\begin{aligned}
h + t + u & = 13 \\
-h + u & = 1 \\
h - t + u & = 3
\end{aligned}
\][/tex]
D.
[tex]\[
\begin{aligned}
h + t + u & = 13 \\
-h + t & = 1 \\
-h - t + u & = 3
\end{aligned}
\][/tex]
The system of equations perfectly matches with Option D.
Therefore, the correct answer is:
[tex]\[
\boxed{1}
\][/tex] (Option D)
