Mathematics, 14.09.2019 07:30 rutza

Solve the linear system by gaussian-jordan elimination. a) x1 + x2 + 2x3 = 8 b) x – y + 2z - w = -1 – x1 – 2x2 + 3x3 = 1 2x + y - 22 – 2w = -2 3x1 – 7x2 + 4x3 = 10 -x + 2y – 4z +w=1 3x – 3w = -3

Answers: 3

Show answers

Answers

Answer from: zakariahassan3482

Solution for the linear system:

a) X_1=3, X_2=1, X_3=2

b) $x= w-1\\y=2z$

z and w are free, meaning that can have any value, for this reason, this system has infinite solutions.

Step-by-step explanation:

Gaussian-Jordan elimination consists of taking an augmented matrix, and transform it into its Row echelon form by means of row operation. For notation, R_i will be the transform column, and r_i the actual one.

Linear System a)

First, you have to convert the system into matrix notation, in this case, column 1 corresponds to variable x_1, column 2 to x_2, column 3 to x_3 and column 4 to the system constants:

$\left[\begin{array}{cccc}1&1&2&8\\-1&-2&3&1\\3&-7&4&10\end{array}\right]$

Transform to 0 every number in the rows under the first row.

Operations:

$R_2=r_1+r_2\\R_3=-3r_1+r_3$

$\left[\begin{array}{cccc}1&1&2&8\\0&-1&5&9\\0&-10&-2&-14\end{array}\right]$

Transform the -1 of the second row into 1

Operations:

R_2=-r_2

$\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&-10&-2&-14\end{array}\right]$

Transform to 0 the numbers under the 1 in the second row

Operations:

R_3=10r_2+r_3

$\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&0&-52&-104\end{array}\right]$

Transform the -52 in the row 3 into 1

Operations:

$R_3=-\frac{1}{52}r_3$

$\left[\begin{array}{cccc}1&1&2&8\\0&1&-5&-9\\0&0&1&2\end{array}\right]$

Now write the matrix like linear equations

$x_1+x_2+2x_3=8\\x_2-5x_3=-9\\x_3=2$

Now that you know the valuo of x_3, you can solve the system starting from the bottom up, then find x_2 and finally x_1

$x_3=2\\x_2=-9+5*2=1\\x_1=8-1-4=3$

Linear System b)

For this system, the process is the same as the above.

Convert the system into matrix form

$\left[\begin{array}{ccccc}1&-1&2&-1&-1\\2&1&-2&-2&-2\\-1&2&-4&1&1\\3&0&0&-3&-3\end{array}\right]$

Operations:

$R_2=-2r_1+r_2\\R_3=r_1+r_3\\R_4=-3r_1+r_4$

$\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&3&-6&0&0\\0&1&-2&0&0\\0&3&-6&0&0\end{array}\right]$

Operations:

$R_2=\frac{1}{3}r_2$

$\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&1&-2&0&0\\0&1&-2&0&0\\0&3&-6&0&0\end{array}\right]$

Operations:

$R_3=-r_2+r_3\\R_4=-3r_2+r_4$

$\left[\begin{array}{ccccc}1&-1&2&-1&-1\\0&1&-2&0&0\\0&0&0&0&0\\0&0&0&0&0\end{array}\right]$

Now you can write the system as equations:

$x-y+2z-w=-1\\y-2z=0$

For w and z there is no unique answer, so the system result is expressed in terms of those variables. This system has infinite solutions.

Solution:

$x= w-1\\y=2z$

z and w are free values.

Answer from: Quest

answer: the 4 kilogram bag

step-by-step explanation: this is because if you divide the number of kilograms into the total it will tell you how much each individual kilogram would be worth. 2 kilos is 7 and change 3 would be around 6 ish and the 4 kilo would be 4.6 and change and the 5 kilos would be like 4.88

Answer from: Quest

point a is the center of the circle that passes through points e, f, and g and the center of the circle that passes through points x, y, and z.

Another question on Mathematics

Mathematics, 21.06.2019 16:00

Quadrilateral wxyz is on a coordinate plane. segment yz is on the line x+y=7 and segment xw is on the line x+y=3. which statement proves how segments yz and xw are related

Answers: 1

Answer

Mathematics, 21.06.2019 19:00

Which values of p and q result in an equation with exactly one solution? px-43=-42x+q choose all answers that apply: a) p=-43 and q=-43 b) p=42 and q=43 c) p=43 and q=−42 d) p=42 and q=42

Answers: 1

Answer

Mathematics, 21.06.2019 22:20

Cone a has a surface area of 384 ft2 and cone b has a surface area of 96 ft? . what is the ratio of their volume?

Answers: 1

Answer

Mathematics, 22.06.2019 02:30

3/4 + p = 1/2? i’m not sure how to solve it

Answers: 1

Answer

You know the right answer?

Solve the linear system by gaussian-jordan elimination. a) x1 + x2 + 2x3 = 8 b) x – y + 2z - w = -1...