# 7.6 Solving systems with gaussian elimination  (Page 5/13)

 Page 5 / 13

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%

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] .

## 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.

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

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

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 $\text{\hspace{0.17em}}{R}_{2}={R}_{2}-9{R}_{1}.\text{\hspace{0.17em}}$ (2) $\text{\hspace{0.17em}}{R}_{2}={R}_{1}-9{R}_{2}.\text{\hspace{0.17em}}$ Then divide row 1 by 9.

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?

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.

## Algebraic

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

$\begin{array}{l}8x-37y=8\\ 2x+12y=3\end{array}$

$\left[\begin{array}{rrrr}\hfill 0& \hfill & \hfill 16& \hfill \\ \hfill 9& \hfill & \hfill -1& \hfill \end{array}|\begin{array}{rr}\hfill & \hfill 4\\ \hfill & \hfill 2\end{array}\right]$

$\left[\begin{array}{rrrrrr}\hfill 1& \hfill & \hfill 5& \hfill & \hfill 8& \hfill \\ \hfill 12& \hfill & \hfill 3& \hfill & \hfill 0& \hfill \\ \hfill 3& \hfill & \hfill 4& \hfill & \hfill 9& \hfill \end{array}|\begin{array}{rr}\hfill & \hfill 16\\ \hfill & \hfill 4\\ \hfill & \hfill -7\end{array}\right]$

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

$\begin{array}{l}-2x+5y=5\\ 6x-18y=26\end{array}$

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
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Abdullahi
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Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
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Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
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Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak