Question: { x + y = 9
x + 2 = 10
y + 2 = 11 }
{ x+y=9
x+z=10
y+z=11 }
1. **State the problem:**
We are given the system of equations:
$$\begin{cases} x + y = 9 \\ x + z = 10 \\ y + z = 11 \end{cases}$$
2. **Goal:**
Find the values of $x$, $y$, and $z$ that satisfy all three equations simultaneously.
3. **Use substitution or elimination:**
From the first equation, express $y$ in terms of $x$:
$$y = 9 - x$$
From the second equation, express $z$ in terms of $x$:
$$z = 10 - x$$
4. **Substitute $y$ and $z$ into the third equation:**
$$y + z = 11$$
Substitute:
$$(9 - x) + (10 - x) = 11$$
5. **Simplify the equation:**
$$9 - x + 10 - x = 11$$
$$19 - 2x = 11$$
6. **Solve for $x$:**
$$19 - 2x = 11$$
Subtract 19 from both sides:
$$19 - 2x - 19 = 11 - 19$$
$$-2x = -8$$
Divide both sides by $-2$:
$$\cancel{-2}x = \cancel{-2} \times 4$$
$$x = 4$$
7. **Find $y$ and $z$ using $x=4$:**
$$y = 9 - x = 9 - 4 = 5$$
$$z = 10 - x = 10 - 4 = 6$$
8. **Check the solution:**
$$y + z = 5 + 6 = 11$$ which matches the third equation.
**Final answer:**
$$x = 4, \quad y = 5, \quad z = 6$$