Solve by elimination:
-2x+2y+3z=0
-2x-y+X=-3
2x+3y+3z=5
-2x-y+X=-3 Huh ?
Oh whoops I meant
-2x-y+z=-3
clearly go for the x's
add 1st and 3rd: 5y + 6z = 5
add 2nd and 3rd : 2y + 4z = 2 or y + 2z = 1
multiply that last one by 5
5y + 10z = 5 , and the other one , as is
5y + 6z = 5
subtract them:
4z = 0
z = 0
sub into y + 2z = 1
y + 0 = 1
y = 1
finally back in -2x+2y+3z=0
-2x + 2 + 0 = 0
-2x = -2
x = 1
To solve the given system of linear equations by elimination, we need to eliminate one variable at a time and then solve for the remaining variables.
Step 1: Eliminate x
Multiply the second equation by 2:
-4x - 2y + 2z = -6
Now, add the first equation to this modified second equation:
-2x + 2y + 3z = 0
+ (-4x - 2y + 2z = -6)
------------------------
-6x + 5z = -6 --------(Equation 1)
Step 2: Eliminate x
Multiply the third equation by 2:
4x + 6y + 6z = 10
Now, add the first equation to this modified third equation:
-2x + 2y + 3z = 0
+ (4x + 6y + 6z = 10)
------------------------
8y + 9z = 10 --------(Equation 2)
We now have two equations with only y and z variables.
Step 3: Solve the equations
Now, we can solve the system of equations formed by Equation 1 and Equation 2.
Multiply Equation 1 by 8 and Equation 2 by 6 to make the coefficients of y in both equations equal and then subtract Equation 2 from Equation 1:
(8*(-6x + 5z)) - (6*(8y + 9z)) = (8*(-6)) - (6*10)
-48x + 40z - 48y - 54z = -48 - 60
-48x - 48y - 14z = -108 --------(Equation 3)
-48x - 48y - 14z = -108
-48x - 48y + 40z = -48
----------------------
-54z = -60
Divide both sides of the equation by -54:
z = -60 / -54
z = 10 / 9 or z = 1.111 (rounded to three decimal places)
Step 4: Substitute the value of z into one of the equations to find y
Substitute z = 10/9 into Equation 1:
-6x + 5z = -6
-6x + 5(10/9) = -6
-6x + 50/9 = -6
-6x = -6 - 50/9
-6x = (-54 - 50)/9
-6x = -104/9
x = (-104/9) / -6
x ≈ 17.333 (rounded to three decimal places)
Step 5: Substitute the values of x and z into one of the equations to find y
Substitute x = 17.333 and z = 10/9 into Equation 2:
8y + 9z = 10
8y + 9(10/9) = 10
8y + 10 = 10
8y = 0
y = 0
Therefore, the solution to the system of equations is:
x ≈ 17.333, y = 0, and z ≈ 1.111