Answer :
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.
[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.
