Of course! Let's solve the system of equations step-by-step.
We have the following system:
[tex]\[
\begin{cases}
2x + 2y = -2 \\
x + y = -1
\end{cases}
\][/tex]
### Step 1: Simplify the equations if possible
First, let's simplify the first equation by dividing every term by 2:
[tex]\[
2x + 2y = -2 \implies x + y = -1
\][/tex]
So, our system of equations is now:
[tex]\[
\begin{cases}
x + y = -1 \\
x + y = -1
\end{cases}
\][/tex]
### Step 2: Analyze the system
Observing both equations, we see that they are identical, meaning they represent the same line. This means that every solution of the first equation is also a solution of the second and vice versa. Hence, we essentially only have one independent equation.
### Step 3: Solve for one variable in terms of the other
Let's solve for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
x + y = -1 \implies y = -1 - x
\][/tex]
### Step 4: Choose any value for [tex]\( x \)[/tex] and find the corresponding [tex]\( y \)[/tex]
Because every point on the line [tex]\( x + y = -1 \)[/tex] is a solution, we can choose any real number for [tex]\( x \)[/tex]. For instance, let's choose [tex]\( x = 0 \)[/tex]:
[tex]\[
y = -1 - 0 = -1
\][/tex]
So one solution is [tex]\((0, -1)\)[/tex].
Alternatively, we could choose [tex]\( x = 1 \)[/tex]:
[tex]\[
y = -1 - 1 = -2
\][/tex]
So another solution is [tex]\((1, -2)\)[/tex].
### General Solution
From the given system, since one equation is simply a multiple of the other, the solutions can be expressed as:
[tex]\[
(x, y) = (a, -1 - a)
\][/tex]
where [tex]\( a \)[/tex] is any real number.
### Conclusion
The system of equations has infinitely many solutions, and the solutions lie on the line [tex]\( x + y = -1 \)[/tex]. Any point [tex]\((a, -1 - a)\)[/tex] where [tex]\( a \)[/tex] is a real number is a solution to the given system.
