To determine which ordered pair [tex]\((x, y)\)[/tex] is the solution to the given system of linear equations, we need to solve the system:
[tex]\[
\begin{cases}
3x + 3y = -3 \\
-2x + y = 2
\end{cases}
\][/tex]
Let's go through the steps to solve this system:
1. Simplify the First Equation:
The first equation can be simplified by dividing each term by 3:
[tex]\[
x + y = -1 \tag{1}
\][/tex]
2. Substitute [tex]\( y \)[/tex] from Equation (1) into the Second Equation:
From Equation (1), we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = -1 - x
\][/tex]
3. Substitute [tex]\( y = -1 - x \)[/tex] into the second equation [tex]\(-2x + y = 2\)[/tex]:
[tex]\[
-2x + (-1 - x) = 2
\][/tex]
Simplify and solve for [tex]\( x \)[/tex]:
[tex]\[
-2x - 1 - x = 2
\][/tex]
[tex]\[
-3x - 1 = 2
\][/tex]
[tex]\[
-3x = 3
\][/tex]
[tex]\[
x = -1
\][/tex]
4. Find [tex]\( y \)[/tex]:
Substitute [tex]\( x = -1 \)[/tex] back into Equation (1):
[tex]\[
y = -1 - (-1)
\][/tex]
[tex]\[
y = -1 + 1
\][/tex]
[tex]\[
y = 0
\][/tex]
Thus, the solution to the system of equations is the ordered pair [tex]\((-1, 0)\)[/tex].
Now, let's check which of the given pairs match this solution:
[tex]\[
\begin{aligned}
&\text{1. } (-1, 0) \quad \text{Yes, it matches.} \\
&\text{2. } (0, -1) \quad \text{No, does not match.} \\
&\text{3. } (1, -2) \quad \text{No, does not match.} \\
&\text{4. } (1, 4) \quad \text{No, does not match.}
\end{aligned}
\][/tex]
So, the ordered pair [tex]\((-1, 0)\)[/tex] is indeed the solution to the given system of linear equations. Therefore, the correct answer is [tex]\((-1, 0)\)[/tex], and it corresponds to the first option provided.
