To solve the given system of equations:
[tex]\[
\begin{cases}
x + 2y = 7 \\
x + y = -10
\end{cases}
\][/tex]
we will use the method of elimination.
Step 1: Set up the equations.
[tex]\[
\begin{array}{c}
(1) \quad x + 2y = 7 \\
(2) \quad x + y = -10
\end{array}
\][/tex]
Step 2: Eliminate one of the variables.
We will subtract equation (2) from equation (1):
[tex]\[
(x + 2y) - (x + y) = 7 - (-10)
\][/tex]
Simplify this equation:
[tex]\[
x + 2y - x - y = 7 + 10
\][/tex]
[tex]\[
y = 17
\][/tex]
Step 3: Substitute the value of [tex]\( y \)[/tex] back into one of the original equations to solve for [tex]\( x \)[/tex].
We will use equation (2):
[tex]\[
x + y = -10
\][/tex]
Substitute [tex]\( y = 17 \)[/tex]:
[tex]\[
x + 17 = -10
\][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[
x = -10 - 17
\][/tex]
[tex]\[
x = -27
\][/tex]
Conclusion:
The solution to the system of equations is:
[tex]\[
\boxed{(-27, 17)}
\][/tex]
So, the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = -27 \)[/tex] and [tex]\( y = 17 \)[/tex].
