1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Oduvanchick [21]
3 years ago
11

Use Gaussian elimination to write each system in triangular form

Mathematics
1 answer:
Feliz [49]3 years ago
7 0

Answer:

To see the steps to the diagonal form see the step-by-step explanation. The solution to the system is x =  -\frac{1}{9}, y= -\frac{1}{9}, z= \frac{4}{9} and w = \frac{7}{9}

Step-by-step explanation:

Gauss elimination method consists in reducing the matrix to a upper triangular one by using three different types of row operations (this is why the method is also called row reduction method). The three elementary row operations are:

  1. Swapping two rows
  2. Multiplying a row by a nonzero number
  3. Adding a multiple of one row to another row

To solve the system using the Gauss elimination method we need to write the augmented matrix of the system. For the given system, this matrix is:

\left[\begin{array}{cccc|c}1 & 1 & 1 & 1 & 1 \\1 & 1 & 0 & -1 & -1 \\-1 & 1 & 1 & 2 & 2 \\1 & 2 & -1 & 1 & 0\end{array}\right]

For this matrix we need to perform the following row operations:

  • R_2 - 1 R_1 \rightarrow R_2 (multiply 1 row by 1 and subtract it from 2 row)
  • R_3 + 1 R_1 \rightarrow R_3 (multiply 1 row by 1 and add it to 3 row)
  • R_4 - 1 R_1 \rightarrow R_4 (multiply 1 row by 1 and subtract it from 4 row)
  • R_2 \leftrightarrow R_3 (interchange the 2 and 3 rows)
  • R_2 / 2 \rightarrow R_2 (divide the 2 row by 2)
  • R_1 - 1 R_2 \rightarrow R_1 (multiply 2 row by 1 and subtract it from 1 row)
  • R_4 - 1 R_2 \rightarrow R_4 (multiply 2 row by 1 and subtract it from 4 row)
  • R_3 \cdot ( -1) \rightarrow R_3 (multiply the 3 row by -1)
  • R_2 - 1 R_3 \rightarrow R_2 (multiply 3 row by 1 and subtract it from 2 row)
  • R_4 + 3 R_3 \rightarrow R_4 (multiply 3 row by 3 and add it to 4 row)
  • R_4 / 4.5 \rightarrow R_4 (divide the 4 row by 4.5)

After this step, the system has an upper triangular form

The triangular matrix looks like:

\left[\begin{array}{cccc|c}1 & 0 & 0 & -0.5 & -0.5  \\0 & 1 & 0 & -0.5 & -0.5\\0 & 0 & 1 & 2 &  2 \\0 & 0 & 0 & 1 &  \frac{7}{9}\end{array}\right]

If you later perform the following operations you can find the solution to the system.

  • R_1 + 0.5 R_4 \rightarrow R_1 (multiply 4 row by 0.5 and add it to 1 row)
  • R_2 + 0.5 R_4 \rightarrow R_2 (multiply 4 row by 0.5 and add it to 2 row)
  • R_3 - 2 R_4 \rightarrow R_3(multiply 4 row by 2 and subtract it from 3 row)

After this operations, the matrix should look like:

\left[\begin{array}{cccc|c}1 & 0 & 0 & 0 & -\frac{1}{9}  \\0 & 1 & 0 & 0 &   -\frac{1}{9}\\0 & 0 & 1 & 0 &  \frac{4}{9} \\0 & 0 & 0 & 1 &  \frac{7}{9}\end{array}\right]

Thus, the solution is:

x =  -\frac{1}{9}, y= -\frac{1}{9}, z= \frac{4}{9} and w = \frac{7}{9}

You might be interested in
What is 2.9 as a mixed number?
lapo4ka [179]

Answer:

2910                                                                                         ….....................….......                                              

3 0
3 years ago
Read 2 more answers
What is the equation of a line that has a slope of 0 and goes through the point (4, 10)?
kompoz [17]

Answer:

y =10

Step-by-step explanation:

The equation of a line in point slope form is expressed as;

y - y0 = m(x-x0)

m is the slope

(x0, y0) is the point on the line

Given

m = 0 and (x0, y0) = (4, 10)

On substituting;

y - 10 = 0(x-4)

y - 10 = 0

y = 0+10

y = 10

Hence the required equation of the line is y =10

4 0
3 years ago
Choose the best description of the following number 18
Mrac [35]

Answer:

4.243

step by step explanation:

5 0
3 years ago
Select the two values of x that are roots of this equation.<br> x2 - 5x + 2 = 0
Margaret [11]

Answer:

C and D

{x}^{2}  - 5x + 2 = 0 \\ x =  \frac{5± \sqrt{17} }{2}

8 0
3 years ago
Read 2 more answers
How do you write 16 tens
Rzqust [24]
16 tens = 16 x 10 = 160 ones
4 0
3 years ago
Read 2 more answers
Other questions:
  • Given the following diagram, find the missing measure.
    13·2 answers
  • Solid:<br> What is the surface area of the pyramid?<br> units
    13·1 answer
  • A tank is measuring 3 in wide by 5 in deep is 10 in tall. If there are 2 cubes with 2 inches sides in the tank, how much water i
    13·1 answer
  • Solve for x.<br><br><br><br> Enter your answer in the box.<br><br> x = <br> °
    9·2 answers
  • CAN SOMEONE ANSWER THIS PLEASEEE AND THANK YOU!
    9·1 answer
  • Help me solve this!!!!!!!!!
    13·1 answer
  • Mmmmmmmmmmmmmmmmmmmmmmmmm
    11·2 answers
  • PLEASEEEEEEEE!!! Graciela was trying to solve the quadratic equation <img src="https://tex.z-dn.net/?f=x%5E2%2B2.5x-1.5%3D0" id=
    12·2 answers
  • PLEASEE HELPP with this problem!!!<br> can you find x and y???
    13·1 answer
  • Divide using long division. State the​ quotient, q(x), and the​ remainder, r(x).
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!