Understanding No Solution Conditions
For a system of linear equations to have no solution, the lines represented by the equations must be parallel. This occurs when the ratios of the coefficients of x and y are equal, but the constant terms are not.
Given Equations
1. kx + 2y = 5
2. 3x + y = 1
Coefficient Comparison
From these equations, we can identify the coefficients:
- For the first equation: A1 = k, B1 = 2, C1 = 5
- For the second equation: A2 = 3, B2 = 1, C2 = 1
Establishing Parallel Lines
To ensure that these lines are parallel, we set up the following condition:
- A1/B1 = A2/B2
This translates to:
- k/2 = 3/1
Solving for k
Cross-multiplying gives us:
- k * 1 = 3 * 2
From this, we find:
- k = 6
Verification of No Solution
Now, we check the constant term condition. For the lines to be parallel but not coincident, we must ensure:
- C1/C2 ≠ A1/A2 (constant terms must not be in the same ratio)
Calculating the ratios:
- 5/1 ≠ 6/3, which simplifies to 5 ≠ 2.
Since this condition holds true, we confirm that the lines are parallel and do not intersect.
Conclusion
The value of k that ensures the system of equations has no solution is:
- k = 6