Answer :
Certainly! To determine which equation can pair with [tex]\( x - y = -2 \)[/tex] to create a consistent and dependent system, we need to analyze each equation in detail:
### Step-by-Step Solution:
1. Initial Equation:
[tex]\[ x - y = -2 \][/tex]
2. Possible Pairing Equations:
- [tex]\( 6x + 2y = 15 \)[/tex]
- [tex]\( -3x + 3y = 6 \)[/tex]
- [tex]\( -8x - 3y = 2 \)[/tex]
- [tex]\( 4x - 4y = 6 \)[/tex]
We need to check the consistency and dependency of the system formed by pairing each possible equation with [tex]\( x - y = -2 \)[/tex].
3. Equation Pair 1:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ 6x + 2y = 15 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{11}{8}, \quad y = \frac{27}{8} \][/tex]
- Verify:
Substitute [tex]\( x = \frac{11}{8} \)[/tex] and [tex]\( y = \frac{27}{8} \)[/tex] back into [tex]\( x - y = -2 \)[/tex]:
[tex]\[ \frac{11}{8} - \frac{27}{8} = -2 \Rightarrow \frac{-16}{8} = -2 \Rightarrow -2 = -2 \][/tex]
The solution satisfies [tex]\( x - y = -2 \)[/tex].
For [tex]\( 6x + 2y = 15 \)[/tex]:
[tex]\[ 6\left(\frac{11}{8}\right) + 2\left(\frac{27}{8}\right) = 15 \Rightarrow \frac{66}{8} + \frac{54}{8} = 15 \Rightarrow \frac{120}{8} = 15 \Rightarrow 15 = 15 \][/tex]
Therefore, this pair is consistent and dependent.
4. Equation Pair 2:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ -3x + 3y = 6 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = y - 2 \][/tex]
- Verify:
Substitute [tex]\( x = y - 2 \)[/tex] into [tex]\( x - y = -2 \)[/tex]:
[tex]\[ (y - 2) - y = -2 \Rightarrow -2 = -2 \][/tex]
The solution satisfies [tex]\( x - y = -2 \)[/tex].
For [tex]\( -3x + 3y = 6 \)[/tex]:
Substitute [tex]\( x = y - 2 \)[/tex]:
[tex]\[ -3(y - 2) + 3y = 6 \Rightarrow -3y + 6 + 3y = 6 \Rightarrow 6 = 6 \][/tex]
This pair is also consistent and dependent.
5. Equation Pair 3:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ -8x - 3y = 2 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = -\frac{8}{11}, \quad y = \frac{14}{11} \][/tex]
- Verify:
Substitute [tex]\( x = -\frac{8}{11} \)[/tex] and [tex]\( y = \frac{14}{11} \)[/tex] back into [tex]\( x - y = -2 \)[/tex]:
[tex]\[ -\frac{8}{11} - \frac{14}{11} = -2 \Rightarrow -\frac{22}{11} = -2 \Rightarrow -2 = -2 \][/tex]
The solution satisfies [tex]\( x - y = -2 \)[/tex].
For [tex]\( -8x - 3y = 2 \)[/tex]:
[tex]\[ -8\left(-\frac{8}{11}\right) - 3\left(\frac{14}{11}\right) = 2 \Rightarrow \frac{64}{11} - \frac{42}{11} = 2 \Rightarrow \frac{22}{11} = 2 \Rightarrow 2 = 2 \][/tex]
This pair is consistent and dependent.
6. Equation Pair 4:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ 4x - 4y = 6 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
The equations result in an empty solution set, meaning they do not have any common solution. Hence, this pair is neither consistent nor dependent.
Among the given equations, the pair with [tex]\( 6x + 2y = 15 \)[/tex] successfully forms a consistent and dependent system with [tex]\( x - y = -2 \)[/tex]. Hence, the correct answer is:
[tex]\[ \boxed{6x + 2y = 15} \][/tex]
### Step-by-Step Solution:
1. Initial Equation:
[tex]\[ x - y = -2 \][/tex]
2. Possible Pairing Equations:
- [tex]\( 6x + 2y = 15 \)[/tex]
- [tex]\( -3x + 3y = 6 \)[/tex]
- [tex]\( -8x - 3y = 2 \)[/tex]
- [tex]\( 4x - 4y = 6 \)[/tex]
We need to check the consistency and dependency of the system formed by pairing each possible equation with [tex]\( x - y = -2 \)[/tex].
3. Equation Pair 1:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ 6x + 2y = 15 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = \frac{11}{8}, \quad y = \frac{27}{8} \][/tex]
- Verify:
Substitute [tex]\( x = \frac{11}{8} \)[/tex] and [tex]\( y = \frac{27}{8} \)[/tex] back into [tex]\( x - y = -2 \)[/tex]:
[tex]\[ \frac{11}{8} - \frac{27}{8} = -2 \Rightarrow \frac{-16}{8} = -2 \Rightarrow -2 = -2 \][/tex]
The solution satisfies [tex]\( x - y = -2 \)[/tex].
For [tex]\( 6x + 2y = 15 \)[/tex]:
[tex]\[ 6\left(\frac{11}{8}\right) + 2\left(\frac{27}{8}\right) = 15 \Rightarrow \frac{66}{8} + \frac{54}{8} = 15 \Rightarrow \frac{120}{8} = 15 \Rightarrow 15 = 15 \][/tex]
Therefore, this pair is consistent and dependent.
4. Equation Pair 2:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ -3x + 3y = 6 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = y - 2 \][/tex]
- Verify:
Substitute [tex]\( x = y - 2 \)[/tex] into [tex]\( x - y = -2 \)[/tex]:
[tex]\[ (y - 2) - y = -2 \Rightarrow -2 = -2 \][/tex]
The solution satisfies [tex]\( x - y = -2 \)[/tex].
For [tex]\( -3x + 3y = 6 \)[/tex]:
Substitute [tex]\( x = y - 2 \)[/tex]:
[tex]\[ -3(y - 2) + 3y = 6 \Rightarrow -3y + 6 + 3y = 6 \Rightarrow 6 = 6 \][/tex]
This pair is also consistent and dependent.
5. Equation Pair 3:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ -8x - 3y = 2 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[ x = -\frac{8}{11}, \quad y = \frac{14}{11} \][/tex]
- Verify:
Substitute [tex]\( x = -\frac{8}{11} \)[/tex] and [tex]\( y = \frac{14}{11} \)[/tex] back into [tex]\( x - y = -2 \)[/tex]:
[tex]\[ -\frac{8}{11} - \frac{14}{11} = -2 \Rightarrow -\frac{22}{11} = -2 \Rightarrow -2 = -2 \][/tex]
The solution satisfies [tex]\( x - y = -2 \)[/tex].
For [tex]\( -8x - 3y = 2 \)[/tex]:
[tex]\[ -8\left(-\frac{8}{11}\right) - 3\left(\frac{14}{11}\right) = 2 \Rightarrow \frac{64}{11} - \frac{42}{11} = 2 \Rightarrow \frac{22}{11} = 2 \Rightarrow 2 = 2 \][/tex]
This pair is consistent and dependent.
6. Equation Pair 4:
[tex]\[ x - y = -2 \][/tex]
[tex]\[ 4x - 4y = 6 \][/tex]
- Solving for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
The equations result in an empty solution set, meaning they do not have any common solution. Hence, this pair is neither consistent nor dependent.
Among the given equations, the pair with [tex]\( 6x + 2y = 15 \)[/tex] successfully forms a consistent and dependent system with [tex]\( x - y = -2 \)[/tex]. Hence, the correct answer is:
[tex]\[ \boxed{6x + 2y = 15} \][/tex]
