Which equation can pair with x-y=-2 to create a consistent and dependent system?
A. 6x+2y= 15
B. -3x+3y=6
C. -8x-3y=2
D. 4x-4y=6
Answer (1)
To find an equation that creates a consistent and dependent system with x − y = − 2 , we look for a scalar multiple of the original equation. The equation − 3 x + 3 y = 6 can be simplified to x − y = − 2 , which is the same as the original equation. Therefore, the answer is − 3 x + 3 y = 6 .
Explanation
Understanding the Problem We are given the equation x − y = − 2 . We need to find another equation that, when paired with the given equation, forms a consistent and dependent system. A consistent and dependent system has infinitely many solutions, meaning the two equations represent the same line. This means the second equation must be a scalar multiple of the first equation.
Finding the Scalar Multiple Multiply the equation x − y = − 2 by a scalar k . The resulting equation is k x − k y = − 2 k . We will check each of the given equations to see if it can be written in the form k x − k y = − 2 k for some value of k .
Analyzing Option 1 Equation 1: 6 x + 2 y = 15 . This cannot be written in the form k x − k y = − 2 k because the coefficients of x and y have opposite signs in the target equation.
Analyzing Option 2 Equation 2: − 3 x + 3 y = 6 . We can rewrite this as − 3 ( x − y ) = 6 , so x − y = − 2 . This is the same as the given equation.
Analyzing Option 3 Equation 3: − 8 x − 3 y = 2 . This cannot be written in the form k x − k y = − 2 k because the coefficients of x and y have the same sign in the target equation.
Analyzing Option 4 Equation 4: 4 x − 4 y = 6 . We can rewrite this as 4 ( x − y ) = 6 , so x − y = 4 6 = 2 3 . This is not the same as the given equation.
Conclusion Therefore, the equation that creates a consistent and dependent system is − 3 x + 3 y = 6 .
Examples
In economics, when analyzing supply and demand, a consistent and dependent system could represent a scenario where the supply and demand curves are essentially the same line. This means that for any given price, the quantity supplied exactly matches the quantity demanded, leading to infinitely many equilibrium points. This situation is rare in real-world markets but serves as a theoretical boundary case for understanding market dynamics and the importance of independent supply and demand relationships.
