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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

An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
Practice Key Terms 7

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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