Question

Find the value of k
3x+2y=2 , (2k-1)x+(k-1)y = 4k+1

Answer 1

To find the value of k in the given system of equations, we can use the method of substitution. By substituting the expressions for x and y into one of the equations and solving for k, we find that k=(-1±sqrt(6))/5.

Step-by-step explanation:

To find the value of k, we can use the method of substitution to solve this system of equations:

1. Start with the first equation: 3x+2y=2
2. Isolate x in terms of y: x=2-(2/3)y
3. Substitute this value of x into the second equation: (2k-1)(2-(2/3)y)+(k-1)y=4k+1
4. Simplify and solve for y: (4k-2)-(4/3)(2k-1)y+(k-1)y=4k+1
5. Combine like terms: (13/3)y=4k+3
6. Substitute this value of y back into the first equation and solve for x: x=2-(2/3)(4k+3)
7. Now we have two equations in terms of k: y=(4k+3)(3/13) and x=2-(2/3)(4k+3)
8. Substitute these expressions for x and y into either equation and solve for k: (2k-1)(2-(2/3)(4k+3))+(k-1)(4k+3)=4k+1
9. Simplify and solve for k: -10k^2+4k-1=0
10. Using the quadratic formula: k=(-4±sqrt(16-4(-10)(-1)))/(2(-10))
11. Simplify: k=(-4±sqrt(96))/(-20)
12. Solve for k: k=(-4±4sqrt(6))/(-20)
13. Final answer: k=(-1±sqrt(6))/5