Final answer:
The system of equations is consistent and independent with the solution X = 16/11 and Y = 24/11.
Step-by-step explanation:
To determine if the system of equations is consistent, independent, dependent, or inconsistent, we need to solve the system of equations.
Given:
4X + Y = 8 ---(1)
X + 3Y = 8 ---(2)
We can solve this system of equations using the method of elimination:
- Multiply equation (2) by 4 to make the coefficients of X in both equations equal.
- 4(X + 3Y) = 4(8)
- 4X + 12Y = 32 ---(3)
- Now, subtract equation (1) from equation (3) to eliminate X.
- (4X + 12Y) - (4X + Y) = 32 - 8
- 11Y = 24
- Y = 24/11
- Substitute the value of Y into any of the original equations to find the value of X.
- Using equation (1): 4X + (24/11) = 8
- 4X = 8 - (24/11)
- 4X = (88 - 24)/11
- 4X = 64/11
- X = (64/11)/4
- X = 64/44 = 16/11
Therefore, the system of equations is consistent and independent. The solution is X = 16/11 and Y = 24/11.