without graphing is each system independent, dependent, or consistent?
-x-3y=0
x-y=4
please someone help me
To determine if the system of equations is independent, dependent, or consistent without graphing, we can use the method of elimination or substitution.
Let's start by using the method of elimination.
Multiply the second equation by 3:
3(x - y) = 3(4)
3x - 3y = 12
Now, you can rewrite the system of equations as:
-x - 3y = 0
3x - 3y = 12
Notice that the coefficients of y in both equations are the same, but the coefficients of x are different. By subtracting the first equation from the second equation, we can eliminate y:
(3x - 3y) - (-x - 3y) = 12 - 0
3x + x = 12
4x = 12
x = 12/4
x = 3
Now that we know x = 3, we can substitute this value back into either of the original equations to find the value of y. Let's choose the second equation:
x - y = 4
3 - y = 4
-y = 4 - 3
-y = 1
y = -1
Therefore, the solution to the system of equations is x = 3 and y = -1.
Since we found a unique solution for both variables, the system of equations is independent and consistent.
To determine whether a system of equations is independent, dependent, or consistent, you need to check if there is a unique solution, infinitely many solutions, or no solution at all.
Let's use the method of substitution to find out.
We have the system of equations:
1) -x - 3y = 0
2) x - y = 4
We can solve equation 2) for x:
x = y + 4
Next, substitute the value of x from equation 2) into equation 1):
-(y + 4) - 3y = 0
Simplifying the equation:
- y - 4 - 3y = 0
-4y - 4 = 0
-4y = 4
y = -1
Now, substitute the value of y into equation 2) to find x:
x = (-1) + 4
x = 3
Therefore, the solution to the system of equations is x = 3 and y = -1. This means there is a unique solution, so the system is independent and consistent.
So, without graphing, we can conclude that the given system of equations is independent and consistent.