To solve this system of equations, we can use the method of substitution or elimination.
1. Method of substitution:
From the second equation, we can isolate y:
-2x + y = 8
y = 8 + 2x
Now substitute this expression for y into the first equation:
3x + 2(8 + 2x) = 2
3x + 16 + 4x = 2
7x + 16 = 2
7x = -14
x = -2
Substitute the value of x back into the second equation to find y:
-2(-2) + y = 8
4 + y = 8
y = 8 - 4
y = 4
Therefore, the solution to the system of equations is x = -2 and y = 4.
2. Method of elimination:
Multiply the second equation by 2 to make the coefficients of y match:
-4x + 2y = 16
Add the two equations together:
(3x + 2y) + (-4x + 2y) = 2 + 16
- x + 4y = 18
Now, multiply the first equation by 4:
4(3x + 2y) = 4(2)
12x + 8y = 8
Now subtract this equation from the previous equation:
(-x + 4y) - (12x + 8y) = 18 - 8
- x + 4y - 12x - 8y = 10 - 8
-13x - 4y = 2
Multiply this equation by -1:
13x + 4y = -2
Add this equation to the previous equation:
(-13x - 4y) + (13x + 4y) = 2 - 2
0 = 0
This equation is always true, which means the two equations are coincident or represent the same line. Therefore, the system of equations has infinitely many solutions.
In summary, the system of equations has infinitely many solutions.