Answer to Question #307683 in Algebra for Ankit

Question

Use row reduction algorithm to solve the following system of equations.

x1 − 7x2 + 6x4 = 5

x3 − 2x4 = −3

−x1 + 7x2 − 4x3 + 2x4 = 7

Answer 1

Let's write down the matrix of our system of linear algebraic equations.

1-7 0 6 | 5

0 0 1 -2 | -3

-1 7 -4 2 | 7

Let's start with the direct move of the Gauss method.

Leave the second line unchanged. That is, zeros located under the main diagonal and on it.

Subtract the last row from the first row of the matrix:

1-7 0 6 | 5

0 0 1 -2 | -3

0 0 -4 4 | 7

Next, multiply the second row by -4 and subtract the last row from it:

1-7 0 6 | 5

0 0 -4 8 | -3

0 0 0 4 | 7

Now we use the inverse of the Gaussian method:

4*x4 = 7, hence x4 = 7/4

We substitute the found values ​​of the variable in the third line:

-4*x3 + 8*x4 = -3

-4 * x3 +14 = -3

-4 * x3 = -17

x3 = 17/4

We substitute the found values ​​in the first line:

x1 - 7*x2 + 6*7/4 = 5

x1 - 7*x2 + 21/2 = 5

x1 - 7*x2 + 10.5 = 5

х1 - 7*х2 = 5 - 10.5

x1 - 7*x2 = -11/2

If we assume that x1 is any rational fraction,

then x2 = 11/14 + x1/7

Thus, the solution of this system of linear algebraic equations can be written as:

x1 - any number from the set Q (rational numbers)

x2 = 11/14 + x1/7

x3 = 17/4

x4 = 7/4