2z - 3y + z = - 19 5x + y - z = - 7 - x + 6y - z = 35
To solve the system of equations, we can use the method of substitution or elimination. Let's use the method of substitution.
From the first equation, we can rewrite it as:
2z + z - 3y = - 19
Combining like terms, we get:
3z - 3y = - 19
Rearranging the equation, we have:
3z = 3y - 19
Dividing both sides by 3, we get:
z = (3y - 19)/3
Substituting this value of z in the second equation, we have:
5x + y - ((3y - 19)/3) = - 7
Multiplying through by 3 to eliminate the fraction, we get:
15x + 3y - (3y - 19) = - 21
Simplifying the equation, we have:
15x + 3y - 3y + 19 = - 21
Combining like terms, we get:
15x + 19 = - 21
Subtracting 19 from both sides, we have:
15x = - 40
Dividing both sides by 15, we get:
x = - 40/15
Simplifying the equation, we have:
x = - 8/3
Now that we have the value of x, we can substitute it into the third equation:
- x + 6y - z = 35
Substituting x = - 8/3, we have:
- (- 8/3) + 6y - z = 35
Simplifying the equation, we get:
8/3 + 6y - z = 35
Rearranging the equation, we have:
6y - z = 35 - 8/3
To eliminate the fraction, we can multiply through by 3:
18y - 3z = 105 - 8
Simplifying the equation, we have:
18y - 3z = 97
Now we have two equations:
3z - 3y = - 19 (equation 1)
18y - 3z = 97 (equation 2)
We can solve this system of equations using the method of elimination.
Multiplying equation 1 by 6 and equation 2 by 1, we have:
18z - 18y = - 114 (equation 3)
18y - 3z = 97 (equation 4)
Adding equation 3 and equation 4, we get:
18z - 18y + 18y - 3z = - 114 + 97
Combining like terms, we have:
15z = - 17
Dividing both sides by 15, we get:
z = - 17/15
Substituting this value of z into equation 1, we have:
3(- 17/15) - 3y = - 19
Multiplying through by 15 to eliminate the fraction, we get:
- 17 - 3y = - 285
Adding 17 to both sides, we have:
- 3y = - 285 + 17
Simplifying the equation, we get:
- 3y = - 268
Dividing both sides by - 3, we get:
y = - 268 / - 3
Simplifying the equation, we get:
y = 268/3
Now we have the values of x, y, and z. Let's substitute them into one of the original equations to check our solution. Let's use the first equation:
2z - 3y + z = - 19
Substituting x = - 8/3, y = 268/3, and z = - 17/15, we have:
2(- 17/15) - 3(268/3) + (- 17/15) = - 19
Multiplying through by 15 to eliminate the fractions, we get:
- 34 - 15(268) - 17 = - 285
Adding 34 and 17 to both sides, we have:
- 15(268) = - 285 + 34 + 17
Simplifying the equation, we get:
- 15(268) = - 234
Dividing both sides by - 15, we get:
268 = 15
Since this is not true, our solution of x = - 8/3, y = 268/3, and z = - 17/15 is incorrect.
Therefore, there is no solution to the system of equations.