To determine which equation can pair with [tex]\( x - y = -2 \)[/tex] to create a consistent and dependent system, we need to understand the criteria for consistency and dependency.
1. Consistent system: A system of equations is consistent if there is at least one solution that satisfies all the equations simultaneously.
2. Dependent system: A system of equations is dependent if it has infinitely many solutions. This typically occurs when the equations describe the same line, or one equation is a multiple of the other.
Here are our given equations:
1. [tex]\( 6x + 2y = 15 \)[/tex]
2. [tex]\( -3x + 3y = 6 \)[/tex]
3. [tex]\( -8x - 3y = 2 \)[/tex]
4. [tex]\( 4x - 4y \pm 6 \)[/tex]
We need to find the pair of equations that form a consistent and dependent system with [tex]\( x - y = -2 \)[/tex].
First, let's evaluate each option.
1. Equation: [tex]\( 6x + 2y = 15 \)[/tex]:
- Simplify [tex]\( x - y = -2 \)[/tex] to express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex]: [tex]\( x = y - 2 \)[/tex].
- Substitute [tex]\( x = y - 2 \)[/tex] into [tex]\( 6x + 2y = 15 \)[/tex]:
[tex]\[
6(y - 2) + 2y = 15 \implies 6y - 12 + 2y = 15 \implies 8y - 12 = 15 \implies 8y = 27 \implies y = \frac{27}{8}.
\][/tex]
- This is a unique solution, making the system consistent but not dependent.
- Result: Not dependent.
2. Equation: [tex]\( -3x + 3y = 6 \)[/tex]:
- Substitute [tex]\( x = y - 2 \)[/tex] into [tex]\( -3x + 3y = 6 \)[/tex]:
[tex]\[
-3(y - 2) + 3y = 6 \implies -3y + 6 + 3y = 6 \implies 6 = 6.
\][/tex]
- This is an identity, meaning the original equation [tex]\( -3x + 3y = 6 \)[/tex] is equivalent to [tex]\( x - y = -2 \)[/tex] when both sides are multiplied by [tex]\(-3\)[/tex].
- Result: Dependent.
3. Equation: [tex]\( -8x - 3y = 2 \)[/tex]:
- Substitute [tex]\( x = y - 2 \)[/tex] into [tex]\( -8x - 3y = 2 \)[/tex]:
[tex]\[
-8(y - 2) - 3y = 2 \implies -8y + 16 - 3y = 2 \implies -11y + 16 = 2 \implies -11y = -14 \implies y = \frac{14}{11}.
\][/tex]
- This is a unique solution, making the system consistent but not dependent.
- Result: Not dependent.
4. Equation: [tex]\( 4x - 4y \pm 6 \)[/tex]:
- We will consider the equation [tex]\( 4x - 4y + 6 = 0 \)[/tex] as it was likely meant to represent an equation precisely.
- Simplify to [tex]\( 4(x - y) + 6 = 0 \)[/tex]:
[tex]\[
4(x - y) + 6 = 0 \implies 4(-2) + 6 = 0 \implies -8 + 6 = -2 \ne 0.
\][/tex]
- This indicates a different process is needed. Consider adjusting terms:
Let's denote [tex]\( 4(x - y) \pm 6 = 0 \)[/tex], where for consistency with [tex]\(x - y = -2\)[/tex], substituting [tex]\(x = y - 2\)[/tex]:
[tex]\[
4(x - y) = -8 \rightarrow -2 + 6 = -2 \rightarrow -8 + 8 = 0
\rightarrow \rightarrow Different terms arise
- Result: Impossible with need of specific term
- Result:
After evaluating the given equations, the equation that can pair with \( x - y = -2 \) to form a consistent and dependent system is:
\[ -3x + 3y = 6 \][/tex]
