Expert: Sherry Wallin - 9/16/2009

Question
How do i solve sytem of equations with three variables?
               x+5y+6z= -10
               x+y+z=2
               x+2y+3z= -3

Answer
Mikey~
    There are a number of ways and not knowing what level of math you have taken I will use the approach most often taught first and that is through substitution and elimination. Label the equations I,II,III
Subtract II from I getting 4y + 5z = -12, call it IV. Do the same with equation I and III, subtract III from I getting 3y + 3z = -7, call it V. Now that you have two new equivalent equations with only 2 variables solve this system of two equations and two unknowns:
4y + 5z = -12  (IV)
3y + 3z = -7   (V)  I choose to multiply IV by -3 and V by 4 because then the y's will go away

-12y -15z = 36
12y + 12z = -28 Now add these two equations and get that -3z = 8 and solve for z getting z = -8/3
substitute back into (V) getting 3y +3(-8/3) = -7 and solve for y getting y = 1/3
Now substitute back into (II) with your new found values for y and z getting:
x + 1/3 -8/3 = 2 finding that x = 13/3. You solution then is {13/3, 1/3, -8/3}, that is, provided I didn't make any arithmetical errors but you can check b plugging back into all 3 original equations and if they are true then the solution is correct.

Math Prof