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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

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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cm
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A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
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Can you compute that for me. Ty
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what is inorganic
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Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
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A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
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you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
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progressive wave
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A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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