Equations solve by matrix method
(1) x-y+3z=2
(2) 2x+y+2z=2
(3) -2x-2y+z=3
1.x=y−3z+2
2.x=−12y−z+1
3.x=−y+12z+−32
Please explain nour
You can check your calculations here:
http://www.wolframalpha.com/input/?i=solve+{{1%2C-1%2C3}%2C{2%2C1%2C2}%2C{-2%2C-2%2C1}}*{{x}%2C{y}%2C{z}}+%3D+{{2}%2C{2}%2C{3}}
sorry, i got that from a culcaltuor
To solve the system of equations using the matrix method, we can represent the equations in matrix form and then use matrix operations to find the solution. Here's how you can do it step by step:
Step 1: Write the augmented matrix
To represent the system of equations in matrix form, we will create an augmented matrix. The augmented matrix includes the coefficients of the variables and the constant terms from each equation. For the given system of equations:
(1) x - y + 3z = 2
(2) 2x + y + 2z = 2
(3) -2x - 2y + z = 3
The augmented matrix will have the following form:
[1 -1 3 | 2]
[2 1 2 | 2]
[-2 -2 1 | 3]
Step 2: Row operations
We will perform row operations to simplify the augmented matrix and bring it to row-echelon form or reduced row-echelon form. The goal is to transform the augmented matrix into upper triangular form or diagonal form.
To do this, we start by performing row operations to create zeroes in the first column below the first row.
First, we multiply the first row by 2 and add it to the second row:
[1 -1 3 | 2]
[0 1 8 | 2]
[-2 -2 1 | 3]
Next, we multiply the first row by -2 and add it to the third row:
[1 -1 3 | 2]
[0 1 8 | 2]
[0 0 7 | -1]
Step 3: Solve for variables
Now that the augmented matrix is in row-echelon form, we can solve for the variables using back substitution.
From the third row, we can deduce that 7z = -1.
So, z = -1/7.
From the second row, we have y + 8z = 2.
Substituting the value of z, we get y + 8(-1/7) = 2.
Simplifying, y - 8/7 = 2.
Therefore, y = 2 + 8/7 = 22/7.
From the first row, we have x - y + 3z = 2.
Substituting the values of y and z, we get x - (22/7) + 3(-1/7) = 2.
Simplifying, x - 22/7 - 3/7 = 2.
Therefore, x = 2 + 22/7 + 3/7 = 36/7.
So, the solution to the system of equations is:
x = 36/7,
y = 22/7,
z = -1/7.
You can check these values by substituting them back into the original equations and verifying that the left-hand side is equal to the right-hand side.