Final answer:
To solve the system of equations, use the method of elimination to eliminate a variable in each step until left with a system of two equations. Then solve for one variable and substitute back to find the values of the other variables.
Step-by-step explanation:
To solve the system of equations:
x - 7y + 3z = 17
5x + 2y - 2z = -57
3x - 10y - z = -11
we can use the method of elimination:
- Multiply the first equation by 5 and the second equation by -3 to eliminate x:
- 5x - 35y + 15z = 85
- -15x - 6y + 6z = 171
- Add the two equations together to eliminate x:
- -41y + 21z = 256
- Multiply the second equation by 3 and the third equation by 5 to eliminate x:
- 15x - 50y - 5z = -165
- 15x - 30y - 3z = 165
- Add the two equations together to eliminate x:
- -80y - 8z = 0
- Now we have a system of two equations:
- -41y + 21z = 256
- -80y - 8z = 0
- Multiply the second equation by -41 to eliminate y:
- 3280y + 328z = 0
- Add the two equations together to eliminate y:
- 349z = 256
- Divide by 349 to solve for z:
- z ≈ 0.735
- Substitute z back into one of the equations to solve for y:
- -80y - 8(0.735) = 0
- y ≈ -0.735
- Finally, substitute the values of y and z back into one of the original equations to solve for x:
- x - 7(-0.735) + 3(0.735) = 17
- x ≈ 10.307