Question

What is the solution of the system of linear equations given below? 4x+3y +2z = 1 5x-2y - z = -1 x-4y + 3z = -7

Answer 1

x = 16/437, y = -63/437, and z = -27/76

Explanation:

To solve the system of linear equations:

4x + 3y + 2z = 1 ---(1)

5x - 2y - z = -1 ---(2)

x - 4y + 3z = -7 ---(3)

We can use any method to solve the system of equations, but here we will use the Gaussian Elimination method to obtain the solution.

Step 1: Write the augmented matrix for the system of equations by combining the coefficients of the variables and the constants as follows:

[4 3 2 | 1]

[5 -2 -1 |-1]

[1 -4 3 |-7]

Step 2: Perform row operations to convert the matrix into row-echelon form.

R2 -> R2 - (5/4)R1 and R3 -> R3 - (1/4)R1

The new matrix is:

[4 3 2 | 1 ]

[0 -23/4 -9/4|-9/4]

[0 -19/4 5/4|-29/4]

Step 3: Continue performing row operations to obtain the matrix in reduced row-echelon form:

R2 -> (-4/23)R2 and R3 -> (-4/19)R3

The new matrix is:

[4 3 2 | 1 ]

[0 1 9/23|-3/23]

[0 0 -76/437|27/437]

Step 4: Convert the matrix back into equations:

4x + 3y + 2z = 1

y + (9/23)z = -3/23

(-76/437)z = 27/437

Step 5: Solve for z:

(-76/437)z = 27/437

z = -27/76

Step 6: Substitute the value of z into the equation y + (9/23)z = -3/23 to obtain y:

y + (9/23)(-27/76) = -3/23

y = -63/437

Step 7: Substitute the values of y and z into the equation 4x + 3y + 2z = 1 to obtain x:

4x + 3(-63/437) + 2(-27/76) = 1

x = 16/437

Therefore, the solution of the system of linear equations is:

x = 16/437, y = -63/437, and z = -27/76

Answer 2

x = -105/227, y = 697/227, z = 122/227

Explanation:

use the method of elimination (also known as Gaussian elimination) or the method of substitution. Here, we will use the method of elimination.

First, we will eliminate the variable x from the second and third equations by multiplying the first equation by 5 and subtracting the result from the second equation, and then multiplying the first equation by 1 and subtracting the result from the third equation. This will give us a system of two equations in two variables:

4x + 3y + 2z = 1 (equation 1)

-17y - 11z = -6 (equation 2')

-16y + z = -8 (equation 3')

Next, we will eliminate y from equation 3' by multiplying equation 2' by -16 and adding the result to equation 3'. This will give us a new equation in the variable z:

4x + 3y + 2z = 1 (equation 1)

-17y - 11z = -6 (equation 2')

-227z = -122 (equation 4')

Finally, we will solve equation 4' for z:

z = 122/227

Substituting this value of z into equation 3', we get:

-16y + (122/227) = -8

Solving for y, we get:

y = 697/227

Substituting these values of y and z into equation 1, we get:

4x + 3(697/227) + 2(122/227) = 1

Solving for x, we get:

x = -105/227

Therefore, the solution to the system of linear equations is:

x = -105/227, y = 697/227, z = 122/227