- The system of linear equations with three variables is:
- x - 3y + 2z = 12
- 2x - 2y + z = 4
- 4x - y + 3z = 9
- To solve, we can use Gaussian elimination to transform the system into triangular form.
- First, eliminate x from the second and third equations.
- Multiply the first equation by 2: 2x - 6y + 4z = 24.
- Subtract this from the second equation: (2x - 2y + z) - (2x - 6y + 4z) = 4 - 24 → 4y - 3z = -20.
- Multiply the first equation by 4: 4x - 12y + 8z = 48.
- Subtract this from the third equation: (4x - y + 3z) - (4x - 12y + 8z) = 9 - 48 → 11y - 5z = -39.
- Now the system is:
- x - 3y + 2z = 12
- 4y - 3z = -20
- 11y - 5z = -39
- Eliminate y from the third equation.
- Multiply the second equation by 11: 44y - 33z = -220.
- Multiply the third equation by 4: 44y - 20z = -156.
- Subtract: (44y - 20z) - (44y - 33z) = -156 - (-220) → 13z = 64 → z = 64/13.
- Substitute z back into the second equation: 4y - 3*(64/13) = -20 → 4y - 192/13 = -20 → 4y = -20 + 192/13 = (-260 + 192)/13 = -68/13 → y = -17/13.
- Substitute y and z into the first equation: x - 3*(-17/13) + 2*(64/13) = 12 → x + 51/13 + 128/13 = 12 → x + 179/13 = 12 → x = 12 - 179/13 = (156 - 179)/13 = -23/13.
- The solution is x = -23/13, y = -17/13, z = 64/13. This is an independent consistent system with a unique solution.
Parent Tip: Review the logic above to help your child master the concept of systems of equations 3 variables worksheet.
