Describe the system 6x-2y=10 and 9x-3y=8 as consistent and independent, consistent and dependent, or inconsistent. Explain.
I don't know how to do this one
Multiply the second equation by 2/3 on each side. I will be happy to critique your thinking on this.
the 2nd equation equals the 1st so there are many solutions to the system so its consistent and dependent.
To determine whether a system of equations is consistent and independent, consistent and dependent, or inconsistent, we need to compare the number of solutions the system has to the number of variables.
Let's start by representing the given system of equations:
Equation 1: 6x - 2y = 10
Equation 2: 9x - 3y = 8
One way to determine the solution type is to compare the slopes of the lines represented by the equations. If the slopes are different, the system is consistent and independent. If the slopes are the same and the y-intercepts are also the same, the system is consistent and dependent. If the slopes are the same, but the y-intercepts are different, the system is inconsistent.
To compare the slopes, let's transform the equations into the slope-intercept form (y = mx + b):
Equation 1: y = 3x - 5 (divide both sides of the equation by -2)
Equation 2: y = 3x - 8/3 (divide both sides of the equation by -3)
Comparing the slopes, we can see that both equations have the same slope, which is 3. This means that the lines represented by the equations are parallel.
However, to confirm whether the system is consistent and dependent or inconsistent, we need to check if the y-intercepts are the same. If they are, the system is consistent and dependent, and if they aren't, the system is inconsistent.
The y-intercept for Equation 1 is -5, while the y-intercept for Equation 2 is -8/3. Since these values are different, we can conclude that the system is inconsistent.
Therefore, the system 6x - 2y = 10 and 9x - 3y = 8 is inconsistent.