Equations as Matrices Practice

1. Write the following system of equations as an augmented matrix.

   Represent each row and column in the order in which the variables and equations appear below.
   

   $\begin{aligned}8x-5y=19\\6x+11y=29\end{aligned}$

2. $\begin{aligned}12x+2y+z+4t&=9\\5x+15y-10z+20t&=-5\end{aligned}$

   Which of the following matrices represents this system?
   

   Choose 1 answer:
   

   1. $\qquad \left[\begin{array} {ccc}12 & 2 & 1 & 4 & -9 \\5 & 15 & -10 & 20 & 5 \end{array} \right]$

   2. $\qquad \left[\begin{array} {ccc}12 & 2 & 4 & 1 & -9 \\5 & 15 & -10 & 20 & 5 \end{array} \right]$

   3. $\qquad \left[\begin{array} {ccc}2 & 12 & 1 & 4 & 9 \\5 & 15 & 10 & 20 & -5 \end{array} \right]$

   4. $\qquad \left[\begin{array} {ccc}12 & 2 & 1 & 4 & 9 \\5 & 15 & -10 & 20 & -5 \end{array} \right]$

3. Write the following system of equations as an augmented matrix.

   Represent each row and column in the order in which the variables and equations appear below.
   

   $\begin{aligned}10x-15y=5\\3x+2y=8\end{aligned}$

4. $\begin{aligned}3x+4y-2t&=-11\\8x-9y-2z+5t&=18\end{aligned}$

   Which of the following matrices represents this system?
   

   Choose 1 answer:
   

   1. $\qquad \left[\begin{array} {ccc}3 & 4 & -2 & 0 & 18 \\8 & -9 & -2 & 5 & -11 \end{array} \right]$

   2. $\qquad \left[\begin{array} {ccc}-11 & 2 & 4 & 0 & -2 \\8 & -9 & -2 & 5 & 18 \end{array} \right]$

   3. $\qquad \left[\begin{array} {ccc}3 & 4 & 0 & -2 & -11 \\8 & -9 & -2 & 5 & 18 \end{array} \right]$

   4. $\qquad \left[\begin{array} {ccc}3 & 4 & 1 & -2 & -11 \\8 & -9 & -2 & 5 & 18 \end{array} \right]$

Source: Khan Academy, https://www.khanacademy.org/math/algebra-home/alg-matrices/alg-representing-systems-with-matrices/e/represent-systems-with-matrices
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.

1. Background

   We can express a system of equations in an augmented matrix in which each row represents one equation. To do this, we write the coefficients of each variable in its own column, and the constants on the right side of the equations in the last column.

   For example, if we are given a system of equations in two variables, then we can express this as an augmented matrix as follows.
   

   $\begin{aligned}ax+by&=m\\cx+dy&=n\end{aligned}$ → $\qquad \left[\begin{array} {ccc}a & b & m \\c & d & n \end{array} \right]$

   Representing the system of equations as a matrix
   

   We are given the system of equations:

   $\begin{aligned}8x-5y=19\\6x+11y=29\end{aligned}$

   First, let's rewrite this system to show the coefficients of each variable.
   

   $\begin{aligned}{8}x+({-5})y&={19}\\{6}x+{11}y&={29}\end{aligned}$
   

   Now, we can write our augmented matrix.

   $\left[\begin{array} {ccc}8 & -5 & 19 \\6 & 11 & 29 \end{array} \right]$
   

   Summary

   Our augmented matrix is:
   

   $\left[\begin{array} {ccc}8 & -5 & 19 \\6 & 11 & 29 \end{array} \right]$
   

   ---

2. Background

   We can express a system of equations in an augmented matrix in which each row represents one equation. To do this, we write the coefficients of each variable in its own column, and the constants on the right side of the equations in the last column.

   For example, if we are given a system of equations in two variables, then we can express this as an augmented matrix as follows.
   

   $\begin{aligned}ax+by&=m\\cx+dy&=n\end{aligned}$ → $\qquad \left[\begin{array} {ccc}a & b & m \\c & d & n \end{array} \right]$

   Representing the system of equations as a matrix
   

   We are given the system of equations:

   $\begin{aligned}12x+2y+z+4t&=9\\5x+15y-10z+20t&=-5\end{aligned}$

   First, let's rewrite this system to show the coefficients of each variable.
   

   $\begin{aligned}{12}x+{2}y+{1}z+{4}t&={9}\\{5}x+{15}y+({-10})z+{20}t&={-5}\end{aligned}$
   

   Now, we can write our augmented matrix.

   $\left[\begin{array} {ccc}12 & 2 & 1 & 4 & 9 \\5 & 15 & -10 & 20 & -5 \end{array} \right]$
   

   Summary

   This is the correct augmented matrix.
   

   $\left[\begin{array} {ccc}12 & 2 & 1 & 4 & 9 \\5 & 15 & -10 & 20 & -5 \end{array} \right]$
   

   ---

3. Background

   We can express a system of equations in an augmented matrix in which each row represents one equation. To do this, we write the coefficients of each variable in its own column, and the constants on the right side of the equations in the last column.

   For example, if we are given a system of equations in two variables, then we can express this as an augmented matrix as follows.
   

   $\begin{aligned}ax+by&=m\\cx+dy&=n\end{aligned}$ → $\qquad \left[\begin{array} {ccc}a & b & m \\c & d & n \end{array} \right]$

   Representing the system of equations as a matrix
   

   We are given the system of equations:

   $\begin{aligned}10x-15y=5\\3x+2y=8\end{aligned}$

   First, let's rewrite this system to show the coefficients of each variable.
   

   $\begin{aligned}{10}x+({-15})y&={5}\\{3}x+{2}y&={8}\end{aligned}$
   

   Now, we can write our augmented matrix.

   $\left[\begin{array} {ccc}10 & -15 & 5 \\3 & 2 & 8 \end{array} \right]$
   

   Summary

   Our augmented matrix is:
   

   $\left[\begin{array} {ccc}10 & -15 & 5 \\3 & 2 & 8 \end{array} \right]$
   

   ---

4. Background

   We can express a system of equations in an augmented matrix in which each row represents one equation. To do this, we write the coefficients of each variable in its own column, and the constants on the right side of the equations in the last column.

   For example, if we are given a system of equations in two variables, then we can express this as an augmented matrix as follows.
   

   $\begin{aligned}ax+by&=m\\cx+dy&=n\end{aligned}$ → $\qquad \left[\begin{array} {ccc}a & b & m \\c & d & n \end{array} \right]$

   Representing the system of equations as a matrix
   

   We are given the system of equations:

   $\begin{aligned}3x+4y-2t&=-11\\8x-9y-2z+5t&=18\end{aligned}$

   First, let's rewrite this system to show the coefficients of each variable.
   

   $\begin{aligned}{3}x+{4}y+{0}z+({-2})t&={-11}\\{8}x+({-9})y+({-2})z+{5}t&={18}\end{aligned}$
   

   Now, we can write our augmented matrix.

   $\left[\begin{array} {ccc}3 & 4 & 0 & -2 & -11 \\8 & -9 & -2 & 5 & 18 \end{array} \right]$
   

   Summary

   This is the correct augmented matrix.
   

   $\left[\begin{array} {ccc}3 & 4 & 0 & -2 & -11 \\8 & -9 & -2 & 5 & 18 \end{array} \right]$