To solve the system of equations:
- (3x - 5y = 2) (Equation 1)
- (-x + 4y = 4) (Equation 2)
We can use the substitution or elimination method. Here, I will use the elimination method.
First, we can multiply Equation 2 by 3 to align the (x) coefficients:
[
3(-x + 4y) = 3(4) \implies -3x + 12y = 12 \quad (Equation 3)
]
Now we have:
- (3x - 5y = 2) (Equation 1)
- (-3x + 12y = 12) (Equation 3)
Now we can add Equation 1 and Equation 3:
[
(3x - 5y) + (-3x + 12y) = 2 + 12
]
This simplifies to:
[
(3x - 3x) + (-5y + 12y) = 14
]
[
0 + 7y = 14
]
Now, isolate (y):
[
7y = 14 \implies y = 2
]
Now that we have (y), we can substitute it back into one of the original equations to find (x). We'll use Equation 2:
[
-x + 4(2) = 4
]
This simplifies to:
[
-x + 8 = 4
]
Now, isolate (x):
[
-x = 4 - 8
]
[
-x = -4 \implies x = 4
]
Thus, the solution to the system of equations is:
[
\boxed{(4, 2)}
]
