Understanding the Problem
The equation we need to analyze is:
- x + y + z = 1
- x + 2y + 4z = kx + 4y + 10z = k^2
We want to find the value of k for which this system of equations has a solution.
Rearranging the Equations
We can simplify the second part of the equation:
1. From x + 2y + 4z = kx + 4y + 10z, we rearrange it to:
- (1 - k)x + (2 - 4)y + (4 - 10)z = 0
- This simplifies to (1 - k)x - 2y - 6z = 0
2. The system of equations now looks like this:
- x + y + z = 1
- (1 - k)x - 2y - 6z = 0
Finding the Value of k
For the system to have solutions, the determinant formed by the coefficients must be zero. Setting up the determinant:
- Coefficient matrix:
| 1 1 1 |
| 1 -2 -6 |
- Setting the determinant to zero gives:
- 1(-2) - 1(-6) = 0
- This leads us to k = 1.
Verifying Solutions
1. For k = 1, the equations become:
- x + y + z = 1
- 0x - 2y - 6z = 0 (which simplifies to y + 3z = 0)
2. We can express y in terms of z:
- y = -3z
3. Substituting back into the first equation:
- x - 3z + z = 1
- x - 2z = 1, giving us x = 1 + 2z.
Thus, for any value of z, we can find corresponding values of x and y. The equations are consistent, confirming k = 1 has solutions.
Conclusion
- The value of k for which the equations have a solution is k = 1.
- The solution set can be expressed in terms of z, leading to infinite solutions.