Practice Problems

Answers

  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]
Back to '9.2: Systems of Linear Equations in Three Variables'