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A small shoe company took out a loan of $1,500,000 to expand their inventory. Part of the money was borrowed at 7%, part was borrowed at 8%, and part was borrowed at 10%. The amount borrowed at 10% was four times the amount borrowed at 7%, and the annual interest on all three loans was $130,500. Use matrices to find the amount borrowed at each rate.

$150,000 at 7%, $750,000 at 8%, $600,000 at 10%

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Access these online resources for additional instruction and practice with solving systems of linear equations using Gaussian elimination.

Key concepts

  • An augmented matrix is one that contains the coefficients and constants of a system of equations. See [link] .
  • A matrix augmented with the constant column can be represented as the original system of equations. See [link] .
  • Row operations include multiplying a row by a constant, adding one row to another row, and interchanging rows.
  • We can use Gaussian elimination to solve a system of equations. See [link] , [link] , and [link] .
  • Row operations are performed on matrices to obtain row-echelon form. See [link] .
  • To solve a system of equations, write it in augmented matrix form. Perform row operations to obtain row-echelon form. Back-substitute to find the solutions. See [link] and [link] .
  • A calculator can be used to solve systems of equations using matrices. See [link] .
  • Many real-world problems can be solved using augmented matrices. See [link] and [link] .

Section exercises

Verbal

Can any system of linear equations be written as an augmented matrix? Explain why or why not. Explain how to write that augmented matrix.

Yes. For each row, the coefficients of the variables are written across the corresponding row, and a vertical bar is placed; then the constants are placed to the right of the vertical bar.

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Can any matrix be written as a system of linear equations? Explain why or why not. Explain how to write that system of equations.

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Is there only one correct method of using row operations on a matrix? Try to explain two different row operations possible to solve the augmented matrix [ 9 3 1 2    |    0 6 ] .

No, there are numerous correct methods of using row operations on a matrix. Two possible ways are the following: (1) Interchange rows 1 and 2. Then R 2 = R 2 −9 R 1 . (2) R 2 = R 1 −9 R 2 . Then divide row 1 by 9.

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Can a matrix whose entry is 0 on the diagonal be solved? Explain why or why not. What would you do to remedy the situation?

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Can a matrix that has 0 entries for an entire row have one solution? Explain why or why not.

No. A matrix with 0 entries for an entire row would have either zero or infinitely many solutions.

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Algebraic

For the following exercises, write the augmented matrix for the linear system.

8 x −37 y = 8 2 x + 12 y = 3

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    16 y = 4 9 x y = 2

[ 0 16 9 −1 | 4 2 ]

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  3 x + 2 y + 10 z = 3 −6 x + 2 y + 5 z = 13               4 x + z = 18

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   x + 5 y + 8 z = 19       12 x + 3 y = 4 3 x + 4 y + 9 z = −7

[ 1 5 8 12 3 0 3 4 9 | 16 4 −7 ]

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6 x + 12 y + 16 z = 4    19 x −5 y + 3 z = −9               x + 2 y = −8

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For the following exercises, write the linear system from the augmented matrix.

[ −2 5 6 −18    |    5 26 ]

−2 x + 5 y = 5 6 x −18 y = 26

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Questions & Answers

find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
Practice Key Terms 7

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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