Answer :

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].