Eliminate y:
- kx + y = 3 ⇒ Equation 1
4x - y = 2 ⇒ Equation 2
-kx : 4x = 3 : 2
-kx (2) = 4x (3)
-2xk = 12x
-2xk/-2x = 12x/-2x
k = - 6
Solve the system, substitute - 6 for k in Equation 1
-(-6)x + y = 3
6x + y = 3
y = -6x + 3 ⇒ Equation 3
Substitute for x by - 6x + 3 for y in Equation 2:
4x - (-6x + 3) = 2
4x + 6x - 3 = 2
10x = 2 + 5
10x/10 = 5/10
x = 1/2
Solve for y, by substituting 1/2 to x in Equation 3:
y = -6x + 3
y = -6(1/2) + 3
y = - 3 + 3
y = 0
The solution to the system is (1/2, 0).
To check, x = 1/2; y = 0
Equation 1:
6x + y = 3
6 (1/2) + 0 = 3
3 + 0 = 3
3 = 3
Equation 2:
4x - y = 2
4 (1/2) - 0 = 2
2 - 0 = 2
2 = 2
Therefore - 6 for k satisfies the system as consistent and independent with only one solution (1/2, 0) which is the point of intersection of the given two equations/graphs.