What is the value of x in the solution of this system?
-3x-y+2z = 7
2y+z= -5
3x+6y= -21
4
2
1
3
thx :)
add the 1st and 3rd
5y + 2z = -14
from the 2nd: z = -2y-5
plug into 5y + 2z = -14
5y + 2(-2y-5) = -14
y = -4
into the 3rd:
3x -24 = -21
3x =3
x = 1
btw: z = 3
To find the value of x in the solution of the system, we need to solve the system of equations.
The given system of equations is:
-3x - y + 2z = 7 ...(1)
2y + z = -5 ...(2)
3x + 6y = -21 ...(3)
To solve the system, we can use either the substitution method or the elimination method. Let's use the elimination method to solve the system:
First, let's eliminate the y variable by multiplying both sides of equation (2) by -1 and adding it to equation (1):
-3x - y + 2z = 7
-(2y + z) = 5
This simplifies to:
-3x - y + 2z = 7
-2y - z = 5
Now, let's eliminate the z variable by multiplying both sides of equation (2) by 2 and adding it to equation (3):
2(2y + z) = 2(-5)
3x + 6y = -21
This simplifies to:
4y + 2z = -10
3x + 6y = -21
We now have a simplified system of equations:
-3x - y + 2z = 7
-2y - z = 5
3x + 6y = -21
4y + 2z = -10
Next, simplify equation (4) by dividing both sides by 2:
2y + z = -5
Now, let's eliminate the z variable by multiplying both sides of equation (2) by 2 and adding it to equation (4):
4(2y + z) = 4(-5)
4y + 2z = -10
This simplifies to:
8y + 4z = -20
4y + 2z = -10
Now, we can write a new system of equations:
-3x - y + 2z = 7
-2y - z = 5
3x + 6y = -21
8y + 4z = -20
4y + 2z = -10
Now, let's eliminate the y variable by multiplying both sides of equation (3) by 2 and adding it to equation (5):
2(3x + 6y) = 2(-21)
4y + 2z = -10
This simplifies to:
6x + 12y = -42
4y + 2z = -10
Now, let's eliminate the y variable by multiplying both sides of equation (5) by -3 and adding it to equation (4):
-3(4y + 2z) = -3(-10)
4y + 2z = -10
This simplifies to:
-12y - 6z = 30
4y + 2z = -10
Now, we have a simplified system of equations:
-3x - y + 2z = 7
-2y - z = 5
6x + 12y = -42
-12y - 6z = 30
4y + 2z = -10
Now, let's eliminate the y variable by multiplying both sides of equation (4) by 2 and adding it to equation (2):
2(-12y - 6z) = 2(30)
-2y - z = 5
This simplifies to:
-24y - 12z = 60
-2y - z = 5
Now, let's eliminate the z variable by multiplying both sides of equation (2) by -12 and adding it to equation (1):
-12(-2y - z) = -12(5)
-3x - y + 2z = 7
This simplifies to:
24y + 12z = -60
-3x - y + 2z = 7
Now, we have a simplified system of equations:
-3x - y + 2z = 7
24y + 12z = -60
-3x - y + 2z = 7
Let's focus on the first and third equations:
-3x - y + 2z = 7
-3x - y + 2z = 7
These two equations are identical, which means that they represent the same line. Therefore, the system of equations is dependent and has infinitely many solutions.
In this case, we can't determine a specific value for x.