Describe the system 6x-2y=10 and 9x-3y=8 as consistent and independent, consistent and dependent, or inconsistent. Explain.
When you multiply the second equation by 2/3, both equations become equal. Therefore there are many solutions. The system is consistent and dependent.
To determine if a system of linear equations is consistent and independent, consistent and dependent, or inconsistent, we need to analyze the relationship between the two equations.
In this case, we have the system of equations:
Equation 1: 6x - 2y = 10
Equation 2: 9x - 3y = 8
To determine if the system is consistent and independent, we need to check if the equations have a unique solution, meaning they intersect at a single point. If they do, the system is consistent and independent.
If the system is consistent and dependent, it means that the equations represent the same line and have infinite solutions. In other words, they are multiples of each other.
If the system is inconsistent, it means that the equations do not intersect and have no common solution.
To analyze the given system, we can perform some algebraic manipulations. A common approach is to check if one equation can be obtained by multiplying the other equation by a constant.
Let's multiply Equation 2 by 2/3:
2/3 * (9x - 3y) = 2/3 * 8
6x - 2y = 16/3
Now, we can see that the left side of the resulting equation matches Equation 1. This means that both equations are equivalent after multiplying Equation 2 by 2/3.
Since the equations are equivalent, they represent the same line. The system has infinitely many solutions because any point on the line is a solution. Therefore, the system is consistent and dependent.