Linear algebra problem

Please solve this linear algebra problem for x, y and z.
Please show all of the steps.

51x + 8y + 37z = 22
99x + 27y + 44z = 83
102x + 96y + 158z = 1096

Let's solve this system of linear equations:

• $$51x+8y+37z=22$$
• $$99x+27y+44z=83$$
• $$102x+96y+158z=1096$$

We can solve the system of equations using elimination.

Steps to solve:

1. Eliminate y: Multiply the top equation by -3 and the second equation by 4. Add the top and bottom equations. $$-153x-24z=-130$$ $$396x+108z=332$$ $$243x+84z=202$$

2. Eliminate z: Multiply the top equation by -1. Add the top and bottom equations. $$243x=-332$$ x=−243332​

3. Substitute x back into the top equation: 51(−243332​)+8y+37z=22 $$-70y+37z=22$$

4. Eliminate y: Multiply the top equation by 19. $$-1330y+703z=418$$

5. Substitute z back into the top equation: −1330y+703(27117517​)=418 y=13555138784​

6. Substitute x and y back into the top equation: 51(−243332​)+13555138784​+37z=22 z=27117517​

Answer:

x=−129571/40665​

y=138784/13555​

z=7517/2711​