can anyone tell me how to use Gaussian Elimination?
3x+2y+4z=19
x+5y+2z=1
Sure! I can help you understand how to use Gaussian Elimination to solve a system of linear equations.
To begin, let's write the system of equations in matrix form:
[3 2 4 | 19]
[1 5 2 | 1]
The goal of Gaussian Elimination is to transform this augmented matrix into row-echelon form or reduced row-echelon form by performing a series of elementary row operations. These operations include:
1. Swapping two rows.
2. Multiplying a row by a non-zero scalar.
3. Adding or subtracting two rows together.
Let's start by eliminating the x-coefficient in the second equation. We can do this by multiplying the first equation by -1 and adding it to the second equation:
[-3 -2 -4 | -19]
[1 5 2 | 1]
Now, let's eliminate the z-coefficient in the second equation. We can do this by multiplying the first equation by -2 and adding it to the second equation:
[-3 -2 -4 | -19]
[0 9 -6 | -37]
Our system of equations has been transformed into row-echelon form. Now, we can use back substitution to solve for the variables. Starting from the bottom row, we can solve for z:
9z - 6 = -37
9z = -31
z = -31/9
Next, substitute the value of z into the second equation to solve for y:
9y - 6z = -37
9y - 6(-31/9) = -37
9y + 62/3 = -37
9y = -37 - 62/3
y = (-37 - 62/3) / 9
Finally, substitute the values of y and z into the first equation to solve for x:
-3x - 2y - 4z = -19
-3x - 2((-37 - 62/3) / 9) - 4(-31/9) = -19
-3x = -19 + 2((-37 - 62/3) / 9) + 4(-31/9)
x = (-19 + 2((-37 - 62/3) / 9) + 4(-31/9)) / -3
This will give you the values of x, y, and z, which are the solution to the system of equations.