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

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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% 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. 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. $\begin{array}{l}-2x+5y=5\\ 6x-18y=26\end{array}$ #### Questions & Answers how do I set up the problem? Harshika Reply what is a solution set? Harshika hello, I am happy to help! Shirley Reply please can go further on polynomials quadratic Abdullahi I need quadratic equation link to Alpa Beta Abdullahi Reply find the value of 2x=32 Felix Reply 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 use the y -intercept and slope to sketch the graph of the equation y=6x Only Reply how do we prove the quadratic formular Seidu Reply hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher Shirley Reply thank you help me with how to prove the quadratic equation Seidu may God blessed u for that. Please I want u to help me in sets. Opoku what is math number Tric Reply 4 Trista x-2y+3z=-3 2x-y+z=7 -x+3y-z=6 Sidiki Reply Need help solving this problem (2/7)^-2 Simone Reply x+2y-z=7 Sidiki what is the coefficient of -4× Mehri Reply -1 Shedrak the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1 Alfred Reply 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?
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