Explain how to solve the following system of equations. What is the solution to the system?
2x+2y+z=-5
3x+4y+2z=0
x+3y+2z=1
I would use elimination.
4x + 4y + 2z = -10 ----- 1st times 2
3x + 4y + 2z = 0 ----- #2 as is
subtract them
x = -10 ------ that was lucky
3x + 4y + 2z = 0 --- #2
x + 3y + 2z = 1 ----#3
subtract them
2x + y = -1
sub x = -10 into that
-20 + y = -1
y = 19
back into #1
2x+2y+z = -5
-20 + 38 + z = -5
z = -23
To solve the given system of equations, we can use the method of elimination or substitution. Let's use the method of elimination:
Step 1: Choose two equations and eliminate one variable.
We can eliminate the variable "x" by multiplying the first equation by 3 and the third equation by 2:
Equation 1 (multiplied by 3): 6x + 6y + 3z = -15
Equation 3 (multiplied by 2): 2x + 6y + 4z = 2
Step 2: Subtract one of the equations from the other to eliminate the common variable.
Subtract Equation 3 from Equation 1:
(6x + 6y + 3z) - (2x + 6y + 4z) = (-15) - (2)
4x - z = -13 ------ Equation 4
Step 3: Repeat Steps 1 and 2 with different combinations of equations to eliminate another variable.
Now, let's eliminate the variable "x" by multiplying the second equation by 2 and subtracting it from the first equation:
Equation 2 (multiplied by 2): 6x + 8y + 4z = 0
(6x + 8y + 4z) - (6x + 6y + 3z) = 0 - (-15)
2y + z = 15 ------ Equation 5
Step 4: Now we have two equations, Equation 4 and Equation 5, with two variables.
Solve these two equations simultaneously to find the values of "y" and "z".
We have:
4x - z = -13 ------------ Equation 4
2y + z = 15 ------------ Equation 5
To eliminate "z", we can multiply Equation 5 by -1 and add it to Equation 4:
(4x - z) + (-2y - z) = (-13) + (-15)
4x - 2y - 2z = -28
2x - y - z = -14 ------------ Equation 6
Step 5: Now, we can solve Equations 4 and 6 simultaneously to find the values of "x" and "y".
We have:
4x - z = -13 ------------ Equation 4
2x - y - z = -14 -------- Equation 6
To eliminate "z", we can multiply Equation 6 by -1 and add it to Equation 4:
(4x - z) + (-2x + y + z) = (-13) + (14)
2x + y = 1 -------------- Equation 7
Step 6: Finally, solve Equations 7 and 5 simultaneously to find the values of "x" and "y".
We have:
2x + y = 1 -------------- Equation 7
2y + z = 15 ------------ Equation 5
To find the values of "x" and "y", we can solve Equation 7 for "x":
2x = 1 - y
x = (1 - y) / 2 ----------- Equation 8
Now, substitute Equation 8 into Equation 5 to get a value for "z":
2y + z = 15
2y + z = 15
z = 15 - 2y
Step 7: Substitute the values of "x", "y", and "z" into one of the three original equations to find the complete solution.
Using the first original equation:
2x + 2y + z = -5
2((1 - y) / 2) + 2y + (15 - 2y) = -5
(1 - y) + 2y + 15 - 2y = -5
1 + 15 = -5
16 = -5
The equation does not hold true, which means there is no solution to the system of equations. The system of equations is inconsistent.