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
DIA [1.3K]
3 years ago
15

Solve each of the following systems by Gauss-Jordan elimination. (b) X1-2x2+ x3- 4x4=1 X1+3x2 + 7x3 + 2x4=2 -12x2-11x3- 16x4 5 (

a) 5x1+2x2 +6x3= 0 -2x1 +x2+3x3 = 0
Mathematics
1 answer:
Len [333]3 years ago
6 0

Answer:

a) The set of solutions is \{(0,-3x_3,x_3): x_3\; \text{es un real}\} y b) the set of solutions is \{(-6,\frac{-41}{17}-\frac{30}{17}x_4 , \frac{37}{17}+\frac{8}{17} x_4 ,x_4): x_4\;\text{es un real}\}.

Step-by-step explanation:

a) Let's first find the echelon form of the matrix \left[\begin{array}{ccc}5&2&6\\-2&1&3\end{array}\right].

  • We add \frac{2}{5} from row 1 to row 2 and we obtain the matrix \left[\begin{array}{ccc}5&2&6\\0&\frac{9}{5} &\frac{27}{5}\end{array}\right]
  • From the previous matrix, we multiply row 1 by \frac{1}{5} and the row 2 by \frac{5}{9} and we obtain the matrix \left[\begin{array}{ccc}1&\frac{2}{5} &\frac{6}{5} \\0&1&3\end{array}\right]. This matrix is the echelon form of the initial matrix.

The system has a free variable (x3).

  • x2+3x3=0, then x2=-3x3
  • 0=x1+\frac{2}{5}x2+\frac{6}{5}x3=

       x1+\frac{2}{5}(-3x3)+\frac{6}{5}x3=

      x1-\frac{6}{5}x3+\frac{6}{5}x3

     then x1=0.

The system has infinite solutions of the form (x1,x2,x3)=(0,-3x3,x3), where x3 is a real number.

b) Let's first find the echelon form of the aumented matrix \left[\begin{array}{ccccc}1&-2&1&-4&1\\1&3&7&2&2\\0&-12&-11&-16&5\end{array}\right].

  • To row 2 we subtract row 1 and we obtain the matrix \left[\begin{array}{ccccc}1&-2&1&-4&1\\0&5&6&6&1\\0&-12&-11&-16&5\end{array}\right]
  • From the previous matrix, we add to row 3, \frac{12}{5} of row 2 and we obtain the matrix \left[\begin{array}{ccccc}1&-2&1&-4&1\\0&5&6&6&1\\0&0&\frac{17}{5}&\frac{-8}{5}&\frac{37}{5}   \end{array}\right].
  • From the previous matrix, we multiply row 2 by \frac{1}{5} and the row 3 by \frac{5}{17} and we obtain the matrix \left[\begin{array}{ccccc}1&-2&1&-4&1\\0&1&\frac{6}{5} &\frac{6}{5}&\frac{1}{5}\\0&0&1&\frac{-8}{17}&\frac{37}{17} \end{array}\right]. This matrix is the echelon form of the initial matrix.

The system has a free variable (x4).

  • x3-\frac{8}{17}x4=\frac{37}{17}, then x3=\frac{37}{17}+ \frac{8}{17}x4.x2+[tex]\frac{6}{5}x3+\frac{6}{5}x4=\frac{1}{5}, x2+\frac{6}{5}(\frac{37}{17}+\frac{8}{17}x4)+[tex]\frac{6}{5}x4=\frac{1}{5}, then

      x2=\frac{-41}{17}-\frac{30}{17}x4.

  • x1-2x2+x3-4x4=1, x1+\frac{82}{17}+\frac{60}{17}x4+\frac{37}{17}+\frac{8}{17}x4-4x4=1, then x1=1-\frac{119}{17}=-6

The system has infinite solutions of the form (x1,x2,x3,x4)=(-6,\frac{-41}{17}-\frac{30}{17}x4,\frac{37}{17}+ \frac{8}{17}x4,x4), where x4 is a real number.

You might be interested in
maggi buys 3/4 pounds of blueberry and uses 3/5 to make a smoothie how many pounds of blueberrys did mggi use to make a smoothie
MrRa [10]

Answer:

0.45pound

Step-by-step explanation:

Since it's given Maggi buys 3/4 pounds of blueberry,it's means 3/4*1 .i.e. 0.75 pound and he uses 3/5 of 0.75 pound then it would be 3/5*0.75 .ie 0.45 pound

3 0
3 years ago
Read 2 more answers
A die is rolled 20 times. given that three of the rolls came up 1, five came up 2, four came up 3, two came up 4, three came up
Serjik [45]

Solution:

Number of times a die is rolled = 20

1 - 3=A

2 - 5=B

3 - 4=C

4 - 2=D

5 - 3=E

6 -  3=F

Total number of arrangements of outcomes , when a dice is rolled 20 times given that 1 appear 3 times, 2 appears 5 times, 3 appear 4 times, 4 appear 2 times , 5 appear three times, and 6 appear 3 times

            = Arrangement of 6 numbers (A,B,C,D,E,F) in 6! ways and then arranging outcomes

= 6! × [ 3! × 5! × 4!×2!×3!×3!]

= 720 × 6×120×24×72→→[Keep in Mind →n!= n (n-1)(n-2)(n-3)........1]

= 895795200  Ways

8 0
3 years ago
Mark and raquel were playing a word game.Raquel played the word math on a game board on a space marked’’triple points’.If her wo
Reika [66]

Answer:

111 points

Step-by-step explanation:

First, thing I like to do is to convert word problems into an eqaution or write down the important information.  So, Math = 37 points and now the points are tripled because of where she placed the word.  Also, tripled is implying that her base score for the word is 3 times the original value.

So, 37*3

= 111 points.

5 0
3 years ago
Read 2 more answers
Randy is painting part of a wall, excluding the two corners shown in white. a gallon of white paint will cover 100 sqaure feet.
uysha [10]

Randy needs to purchase 1.49 Gallons of paint to cover 100 sq. feet wall.

What is the area of rectangle?

The area of a rectangle is the product of its length and width. That is,       A = l x w where l is the length and w is the width of the rectangle.

Here, Required Paint = 20 X \frac{37}{4} - 2 X \frac{1}{2}X6 X 6\\

                                   = 5 X 37  - 36

                                   = 185 - 36

                                   = 149

                                   = 1.49 gallons

Thus, Randy needs to purchase 1.49 Gallons of paint to cover 100 sq. feet wall.

Learn more about Area of Rectangle from :

brainly.com/question/20693059

#SPJ1

4 0
2 years ago
The length of the carpet is 8 yards 9 inches.
zzz [600]

Multiply 8 by 9, this equals 72

Your answer is 72 Inches

7 0
3 years ago
Read 2 more answers
Other questions:
  • MATH :((<br>how do the chart??<br><img src="https://tex.z-dn.net/?f=%20-%20x%20%20%5E%7B2%7D%20%20-%209%20%5Cgeqslant%200" id="T
    13·1 answer
  • Point R is located at (3, 2) and point S is located at (8, 15) . What are the coordinates of the point that partitions the direc
    7·1 answer
  • Guys I really need help with these too thanks u before!!!!
    6·1 answer
  • what s the probality that the first card is a king and the second card is a king if the smapling is done withot replacement
    13·1 answer
  • In the following series of numbers, please identify the lowest number? 1.7 7 0.7 0.37 17
    6·1 answer
  • How to eliminate square roots
    5·2 answers
  • Han used 1 1/2 gallons of paint to paint 3/5 of a shed. How many gallons of paint will it take to paint the whole shed? Write a
    9·1 answer
  • Which of the following is the solution set to the equation -4(x+1)^2 -3=- 51
    6·1 answer
  • Julie walks 10 dogs a week and gets paid $4.25 for each dog. How much money does
    5·2 answers
  • A) x²-4x+3=0 prove in factorization form<br>​
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!