without graphing classify each system as independent, dependent, or inconsistent.
-2x-y=9
3x-4y=-8
They are independent because the slopes are different. The slope is the ratio of the coefficients of the x and y terms. The lines that correspond to these equations must therefore intersect at a solution point.
To classify the system as independent, dependent, or inconsistent without graphing, you can use the method of solving the system of equations using elimination or substitution.
Using elimination method:
1. Multiply the first equation by 3 and the second equation by 2 to make the coefficients of x in both equations the same:
-6x - 3y = 27
6x - 8y = -16
2. Add the two equations to eliminate x:
-11y = 11
3. Solve for y:
y = -1
4. Substitute the value of y back into one of the original equations to solve for x:
-2x - (-1) = 9
-2x + 1 = 9
-2x = 8
x = -4
The solution to the system is x = -4 and y = -1.
Since the solution is a unique point, the system is classified as an independent system.
To classify the given system of equations as independent, dependent, or inconsistent, we can use the concept of consistent and inconsistent systems.
In a consistent system of linear equations, there is at least one solution that satisfies all the given equations. Independent systems have exactly one solution, while dependent systems have infinitely many solutions. In an inconsistent system, there is no solution that satisfies all the equations simultaneously.
To determine the classification without graphing, we can solve the given system using a method such as substitution or elimination.
Let's use the elimination method to solve the system:
Step 1: Multiply the first equation by 4 to eliminate the variable x.
-2x - y = 9 -> -8x - 4y = 36
Step 2: Multiply the second equation by 2 to eliminate the variable x.
3x - 4y = -8 -> 6x - 8y = -16
Now we have the following system of equations:
-8x - 4y = 36
6x - 8y = -16
Step 3: Add the two equations to eliminate the variable x.
(-8x + 6x) + (-4y - 8y) = 36 + (-16)
-2x - 12y = 20
Simplifying further, we have:
-2x - 12y = 20
At this point, we can see that the third equation we obtained is different from the previous two equations. It does not match either of the original equations.
So, we can conclude that the given system of equations is inconsistent since we obtain an equation that does not have any connection/relation to the original equations. In other words, the system has no solution that satisfies all the equations simultaneously.
Therefore, the classification of the given system is: inconsistent.