4x−y+2z=11
x+2y−z=−1
2x+2y−3z=−1
can someone help me find x,y,z, I have tried isolation but i cant seem to get it
never mind guys i got it cuz im smart and i never give up.
I would use elimination, especially after looking at the y terms
#3 - #2: x - 2z= 0 , so we already know z = 2y
#1 times 2 : 8x - 2y + 4z = 22
original #2 as is: x + 2y - z = -1
add them:
9x + 3z = 21
or
3x + z = 7
double this: 6x + 2z = 14
using with : x - 2z = 0
add them
7x = 14
x = 2
in 3x+z=7 ----> 6+z=7
z = 1
in the original #2:
x+2y-z=-1
2+2y-1=-1
2y=-2
y=-1
x=2 , y=-1, z=1
The above solution is not unique.
Other than choosing the y's, there was really no plan, you just sort of go with the flow.
To solve the system of equations, you can use a method called "Gaussian elimination" or "row reduction." This involves performing operations on the equations to get them into a simplified form that reveals the values of x, y, and z.
Let's start by expressing the system of equations in matrix form:
[A] * [X] = [B]
where [A] is the coefficient matrix, [X] is the column matrix of the variables (x, y, z), and [B] is the constant matrix.
In our case, [A] would be:
| 4 -1 2 |
| 1 2 -1 |
| 2 2 -3 |
[X] would be:
| x |
| y |
| z |
And [B] would be:
| 11 |
| -1 |
| -1 |
Now, let's perform row operations to simplify the system:
1. Multiply Equation 1 by 1/4 to make the leading coefficient of x equal to 1:
| 4 -1 2 | | 11 |
| 1 2 -1 | -> | -1 |
| 2 2 -3 | | -1 |
2. Replace Equation 2 with Equation 2 - Equation 1:
| 4 -1 2 | | 11 |
| 0 9 -3 | -> | -8 |
| 2 2 -3 | | -1 |
3. Replace Equation 3 with Equation 3 - 2 * Equation 1:
| 4 -1 2 | | 11 |
| 0 9 -3 | -> | -8 |
| 0 4 -7 | | -13 |
4. Multiply Equation 2 by 1/9 to make the leading coefficient of y equal to 1:
| 4 -1 2 | | 11 |
| 0 9 -3 | -> | -8 |
| 0 4 -7 | | -13 |
5. Replace Equation 3 with Equation 3 - (4/9) * Equation 2:
| 4 -1 2 | | 11 |
| 0 9 -3 | -> | -8 |
| 0 0 -1 | | -5 |
Now, we have simplified the system to the form [A] * [X] = [B], where A is an upper triangular matrix.
To solve for z, we can directly substitute z = -5 into the third equation (0 * x + 0 * y - 1 * z = -5). So, z = -5.
Substituting z = -5 into the second equation (0 * x + 9 * y - 3 * (-5) = -8), we can solve for y. Therefore, 9y - 15 = -8, yielding y = 1.
Finally, substituting z = -5 and y = 1 into the first equation (4 * x - 1 * 1 + 2 * (-5) = 11), we can solve for x. Hence, 4x - 1 - 10 = 11, giving us x = 6.
So, the solution to the system of equations is x = 6, y = 1, and z = -5.