6x+5y-6z=-9
-x-y+3z=10
-6x-y+z=23
multiply the 2nd eqn. by 6
add 1st eqn to 2nd eqn, and to 3rd eqn
... this gives two eqns in y and z (x has been eliminated)
solve for y (or z) , then substitute back to find all unknowns
Let's go for the y's
the 2nd: -x -y + 3z = 10
the 3rd: -6x - y + z = 23
subtract them:
5x + 2z = -13 **
the first as is: 6x + 5y - 6z = -9
5 times the 2nd: -5x -5y + 15z = 50
add those ...
x + 9z = 41 or x = 41- 9z ***
sub *** into **
5(41 - 9z) + 2z = -13
205 - 45z + 2z = -13
-43z = -218
z = 218/43
back in **
x = 41 - 9z = 41 - 9(218/43) = ....
now put that back into the original 2nd equation to find y
To solve this system of equations, we can use the method of elimination or substitution. I will explain both methods and you can choose the one you prefer to solve the given system of equations.
Method 1: Elimination
1. Multiply the second equation by 6 and add it to the third equation, which will eliminate the term "-6x" from the third equation.
(-x-y+3z) * 6 + (-6x-y+z) = 10 * 6 + 23
-6x - 6y + 18z - 6x - y + z = 60 + 23
-12x - 7y + 19z = 83
2. Multiply the first equation by 6 and add it to the new third equation to eliminate "6x" from the third equation.
(6x+5y-6z) * 6 + (-12x - 7y + 19z) = -9 * 6 + 83
36x + 30y - 36z - 12x - 7y + 19z = -54 + 83
24x + 23y - 17z = 29
3. Now we have a system of two equations with two variables:
24x + 23y - 17z = 29 --(1)
-12x - 7y + 19z = 83 --(2)
4. Multiply equation (1) by 12 to eliminate "12x" from equation (2):
(24x + 23y - 17z) * 12 + (-12x - 7y + 19z) = 29 * 12 + 83
288x + 276y - 204z - 12x - 7y + 19z = 348 + 83
276x + 269y - 185z = 431
5. Add equation (2) multiplied by 276 to equation (3) to eliminate "276x":
(276x + 269y - 185z) + (276x + 269y - 185z) = 431 + (276 * 83)
552x + 538y - 370z = 431 + 22908
552x + 538y - 370z = 23339
6. Now we have:
552x + 538y - 370z = 23339 --(3)
276x + 269y - 185z = 431 --(4)
7. Solve equations (3) and (4) to find the values of x, y, and z.
Method 2: Substitution
1. Solve one equation for one variable in terms of the other variables and substitute it into the other equations.
From the first equation: 6x + 5y - 6z = -9 --> x = (-5y + 6z - 9)/6
Substitute this value of x in the second equation:
-(-5y + 6z - 9)/6 - y + 3z = 10
(5y - 6z + 9)/6 - y + 3z = 10
2. Solve the resulting equation for one variable (preferably the one with the least number of fractions).
(5y - 6z + 9)/6 - y + 3z = 10
5y - 6z + 9 - 6y + 18z = 60
-y + 12z = 51
3. Now we have two equations:
-y + 12z = 51 --(2)
-6x - y + z = 23 --(5)
4. Solve equations (2) and (5) to find the values of x, y, and z.
Both methods will lead to the same solution. You can choose the one that you find more comfortable to use.