What is the solution?
2x-3y+z=-19
5x+y-z=-7
-x+6y-z=35
substitution:
z = -19-2x+3y
so use that to get
5x+y-(-19-2x+3y) = -7
-x+6y-(-19-2x+3y) = 35
or
7x-2y = -26
x+3y = 16
Now you have x = 16-3y, so that gives
7(16-3y)-2y = -26
-23y = -138
y = 6
so, x=16-3y = 16-3*6 = -2
so, z = -19-2x+3y = -19-2(-2)+3*6 = 3
so, the solution is (-2,6,3)
elimination:
2x-3y+z=-19
5x+y-z=-7
-x+6y-z=35
add #1 and #2, then #1 and #3
7x-2y = -26
x+3y = 16
add 3*#1 to 2*#2
21x-6y = -78
2x+6y = 32
or
23x = -46
x = -2 as above
To find the solution to the given system of equations, you can use the method of elimination or substitution. Let's use the method of elimination to solve these equations.
Step 1: Multiply the second equation by 2 and the third equation by 5 to ensure that the coefficients of z will cancel out when we eliminate it.
2x - 3y + z = -19 (Equation 1)
10x + 2y - 2z = -14 (Equation 2)
-5x + 30y - 5z = 175 (Equation 3)
Step 2: Add Equation 2 and Equation 3 to eliminate z.
10x + 2y - 2z + (-5x + 30y - 5z) = -14 + 175
5x + 32y - 7z = 161 (Equation 4)
Step 3: Add Equation 1 and Equation 4 to eliminate z.
2x - 3y + z + (5x + 32y - 7z) = -19 + 161
7x + 29y = 142 (Equation 5)
Now we have a new equation, Equation 5, that only involves x and y.
Step 4: Solve Equation 5 for one variable (either x or y).
Let's solve Equation 5 for x:
7x = 142 - 29y
x = (142 - 29y) / 7 (Equation 6)
Step 5: Substitute the expression for x from Equation 6 into one of the original equations to solve for the remaining variable (either x or y).
Let's substitute Equation 6 into Equation 1:
2((142 - 29y) / 7) - 3y + z = -19
284 - 58y - 21y + 7z = -133
-79y + 7z = -417 (Equation 7)
Step 6: Solve Equation 7 for one variable (either y or z).
Let's solve Equation 7 for y:
-79y = -417 - 7z
y = (-417 - 7z) / -79 (Equation 8)
Step 7: Substitute the expression for y from Equation 8 into one of the original equations to solve for the remaining variable (either x or z).
Let's substitute Equation 8 into Equation 1:
2x - 3((-417 - 7z) / -79) + z = -19
2x + (1251 + 21z) / 79 + z = -19
2x + (1251 + 21z + 79z) / 79 = -19
2x + (1251 + 100z) / 79 = -19
2x = (-1504z - 1272) / 79 - 19
2x = (-1504z - 1272 - 79*19) / 79
x = (-1504z - 1272 - 79*19) / (2*79) (Equation 9)
Step 8: The solution to the system of equations is the three variables x, y, and z, obtained from Equations 6, 8, and 9, respectively.