Which system of equations is consistent and dependent?
$\left\{\begin{array}{l}3 x+2 y=3 \\ 8 x+4 y=6\end{array}\right.$
$\left\{\begin{array}{l}4 x+6 y=2 \\ 4 x+6 y=1\end{array}\right.$
$\left\{\begin{array}{l}2 x+3 y=-8 \\ 2 x+3 y=-12\end{array}\right.$
$\left\{\begin{array}{l}5 x+4 y=-30 \\ 3 x-8 y=-18\end{array}\right.$
Answer (1)
A consistent and dependent system has infinitely many solutions, meaning the equations represent the same line.
System 1, 2, and 3 are not consistent and dependent because they do not represent the same line.
System 4 is not consistent and dependent because it has a unique solution.
Therefore, none of the systems are consistent and dependent: None .
Explanation
Understanding Consistent and Dependent Systems We need to determine which of the given systems of equations is consistent and dependent. A consistent and dependent system has infinitely many solutions, meaning the two equations represent the same line. This implies that one equation is a scalar multiple of the other.
Analyzing Each System Let's analyze each system:
System 1: { 3 x + 2 y = 3 8 x + 4 y = 6
We can try to multiply the first equation by a constant to see if it matches the second equation. If we multiply the first equation by 3 8 , we get 8 x + 3 16 y = 8 , which is not the same as the second equation. If we multiply the first equation by 2, we get 6 x + 4 y = 6 , which is also not the same as the second equation. Therefore, System 1 is not consistent and dependent.
System 2: { 4 x + 6 y = 2 4 x + 6 y = 1
Here, the left-hand sides of the equations are the same, but the right-hand sides are different. This means the system is inconsistent and has no solution. Therefore, System 2 is not consistent and dependent.
System 3: { 2 x + 3 y = − 8 2 x + 3 y = − 12
Similar to System 2, the left-hand sides are the same, but the right-hand sides are different. This system is also inconsistent and has no solution. Therefore, System 3 is not consistent and dependent.
System 4: { 5 x + 4 y = − 30 3 x − 8 y = − 18
These equations are independent and will have a unique solution. We can solve this system using substitution or elimination to find a unique solution. Therefore, System 4 is not consistent and dependent.
Conclusion After analyzing all the systems, we find that none of them are consistent and dependent.
Final Answer Based on the analysis, none of the provided systems of equations are consistent and dependent.
Examples
In economics, understanding systems of equations helps in modeling supply and demand curves. A consistent and dependent system would imply that the supply and demand curves are essentially the same, leading to infinitely many equilibrium points. This scenario is rare but could occur if two different models describe the exact same market conditions. Analyzing such systems helps economists understand the relationships between different economic variables and make predictions about market behavior.
