Identify the system as consistent, inconsistent or dependent. Explain your choice.
1.) 3x - 2y = 12
6x - 4y = 24
2.) x= -5
y= 4
1. dependent. One equation can be derived from the other.
2. consistent. Both equations can be true
To determine whether a system of equations is consistent, inconsistent, or dependent, we need to look at the solutions of the system.
1.) 3x - 2y = 12
6x - 4y = 24
We can start by rearranging the equations to solve for x:
First, we divide the second equation by 6 to simplify it:
6x - 4y = 24
Dividing by 6, we get:
x - 2/3y = 4
Now, we have the following system of equations:
3x - 2y = 12
x - 2/3y = 4
To determine consistency, we need to compare the slopes. The coefficients of x in both equations are the same, which means the lines are parallel. Since parallel lines never intersect, the system is inconsistent. There are no solutions that satisfy both equations.
2.) x = -5
y = 4
In this case, we have two equations that are already solved for x and y. By substituting these values into the original equations, we can check if they satisfy both equations:
Substituting x = -5 and y = 4 into the first equation:
3x - 2y = 12
3(-5) - 2(4) = -15 - 8 = -23 ≠ 12
Since the substitution does not satisfy the first equation, the system is inconsistent. There are no values of x and y that satisfy both equations.
Therefore, the first system is inconsistent, and the second system is also inconsistent.