I need help I don't know how to solve this system
Solve the following system. Show all your work
3x-2y+2z= 30
-x+3y-4z= -33
2x-4y+3z= 42
what makes it especially difficult? You must have solved similar systems already.
2(1st)+(2nd) and 3(1st)-2(3rd):
5x - y = 27
5x + 2y = 6
Now it gets easier:
3y = -21
y = -7
so,
x = 4
z = 2
every time i try adding two equations that could make it simpler ex; by adding and crossing out variables. i just could never get it right and still have those three same variables thx for the help
Yeah, it takes a bit of practice. Look for LCMs to get rid of stuff. In this case, since 3*2=6, we had to take 3 times the 1st to get 6 and 2 times the 3rd to get 6.
Usually the errors come in watching the other coefficients. In the beginning, it helps to write everything down so details don't get lost.
23
To solve this system of linear equations, you can use the method of elimination or substitution. I'll explain the elimination method, which involves eliminating one variable at a time by adding or subtracting the equations.
Step 1: Start by eliminating the variable 'x' from the equations. Multiply the second equation by 3 and the first equation by 1, which gives:
-3x + 9y - 12z = -99
3x - 2y + 2z = 30
Adding these two equations eliminates 'x', and we get:
7y - 10z = -69 --(Equation 3)
Step 2: Next, eliminate the variable 'x' again by multiplying the third equation by 2 and the first equation by 2:
4x - 8y + 6z = 84
6x - 4y + 4z = 60
Adding these two equations eliminates 'x' once more, resulting in:
-12y + 10z = -24 --(Equation 4)
Step 3: Now we have a system of two equations with two variables, 'y' and 'z':
Equation 3: 7y - 10z = -69
Equation 4: -12y + 10z = -24
To eliminate 'z', we'll add the two equations:
7y - 10z + (-12y + 10z) = -69 + (-24)
This simplifies to:
-5y = -93
Step 4: Solve for 'y' by dividing both sides of the equation by -5:
y = (-93) / (-5)
y = 18.6
Step 5: Substitute the value of 'y' back into either Equation 3 or Equation 4 to solve for 'z'. Using Equation 3:
7(18.6) - 10z = -69
118.2 - 10z = -69
-10z = -69 - 118.2
-10z = -187.2
Divide both sides of the equation by -10:
z = (-187.2) / (-10)
z = 18.72
Step 6: Finally, substitute the values of 'y' and 'z' back into any of the original equations to solve for 'x'. Let's use the first equation:
3x - 2(18.6) + 2(18.72) = 30
3x - 37.2 + 37.44 = 30
3x + 0.24 = 30
3x = 30 - 0.24
3x = 29.76
Divide both sides of the equation by 3:
x = 29.76 / 3
x = 9.92
Therefore, the solution to the system of equations is x = 9.92, y = 18.6, and z = 18.72.