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Section 1.2 Finding solutions to systems of linear equations
One of our goals will be to use algbra to find solutions to our systems of equations.
Activity 1.2.0.1 .
Use algebra to show that the following two equations cannot be true at the same time.
\begin{align*}
-x_1+2x_2 &= 5\\
2x_1-4x_2 &= 6
\end{align*}
You are showing that the equations are inconsistent . What contradiction do you reach?
Activity 1.2.0.2 .
If a solutions exists, the system is consistent . Consider the following linear system.
\begin{align*}
-x_1+2x_2 &= -3\\
2x_1-4x_2 &= 6
\end{align*}
(a) Find three different solutions for this system.
(b) Let \(x_2=a\) where \(a\) is an arbitrary real number, then find an expression for \(x_1\) in terms of \(a\text{.}\) Use this to write the solution set \(\setBuilder
{
\left[\begin{array}{c}
\unknown \\
a
\end{array}\right]
}{
a \in \IR
}\) for the linear system.
Activity 1.2.0.3 .
Consider the following linear system.
\begin{alignat*}{5}
x_1 &\,+\,& 2x_2 &\, \,& &\,-\,& x_4 &\,=\,& 3\\
&\, \,& &\, \,& x_3 &\,+\,& 4x_4 &\,=\,& -2
\end{alignat*}
Describe the solution set
\begin{equation*}
\setBuilder
{
\left[\begin{array}{c}
\unknown \\
a \\
\unknown \\
b
\end{array}\right]
}{
a,b \in \IR
}
\end{equation*}
to the linear system by setting \(x_2=a\) and \(x_4=b\text{,}\) and then solving for \(x_1\) and \(x_3\text{.}\)
We can use an augmented matrix to represent the system of equations.
Definition 1.2.0.1 .
Two systems of linear equations (and their corresponding augmented matrices) are said to be equivalent if they have the same solution set.
Activity 1.2.0.4 .
Following are seven procedures used to manipulate an augmented matrix. If the procedure results in an equivalent augmented matrix (preseves the corresponding solution set), label the procedure as valid . If the procedure might change the solution set of the corresponding linear system, label it as invalid .
Swap two rows.
Swap two columns.
Add a constant to every term in a row.
Multiply a row by a nonzero constant.
Add a constant multiple of one row to another row.
Replace a column with zeros.
Replace a row with zeros.
Two matrices are row equivalent if we can use a series of valid row operation to get from one matrix to the next.
Activity 1.2.0.5 .
(a)
Suppose that you have a system of linear equations in the unknowns \(x\) and \(y\) whose augmented matrix is row equivalent to
\begin{equation*}
\left[
\begin{array}{rr|r}
1 \amp 0 \amp 3 \\
0 \amp 1 \amp 0 \\
0 \amp 0 \amp 0 \\
\end{array}
\right].
\end{equation*}
Write the system of linear equations corresponding to the augmented matrix. Then describe the solution set of the system of equations in as much detail as you can.
(b)
Suppose that you have a system of linear equations in the unknowns \(x\) and \(y\) whose augmented matrix is row equivalent to
\begin{equation*}
\left[
\begin{array}{rr|r}
1 \amp 0 \amp 3 \\
0 \amp 1 \amp 0 \\
0 \amp 0 \amp 1 \\
\end{array}
\right].
\end{equation*}
Write the system of linear equations corresponding to the augmented matrix. Then describe the solution set of the system of equations in as much detail as you can.
Activity 1.2.0.6 .
Consider the following (equivalent) linear systems.
A)
\begin{alignat*}{4}
x &\,+\,& 2y &\,+\,& z &\,=\,& 3 \\
-x &\,-\,& y &\,+\,& z &\,=\,& 1 \\
2x &\,+\,& 5y &\,+\,& 3z &\,=\,& 7
\end{alignat*}
B)
\begin{alignat*}{4}
2x &\,+\,& 5y &\,+\,& 3z &\,=\,& 7 \\
-x &\,-\,& y &\,+\,& z &\,=\,& 1 \\
x &\,+\,& 2y &\,+\,& z &\,=\,& 3
\end{alignat*}
C)
\begin{alignat*}{4}
x & & &\,-\,& z &\,=\,& 1 \\
& & y &\,+\,& 2z &\,=\,& 4 \\
& & y &\,+\,& z &\,=\,& 1
\end{alignat*}
D)
\begin{alignat*}{4}
x &\,+\,& 2y &\,+\,& z &\,=\,& 3 \\
& & y &\,+\,& 2z &\,=\,& 4 \\
2x &\,+\,& 5y &\,+\,& 3z &\,=\,& 7
\end{alignat*}
E)
\begin{alignat*}{4}
x & & &\,-\,& z &\,=\,& 1 \\
& & y &\,+\,& z &\,=\,& 1 \\
& & & & z &\,=\,& 3
\end{alignat*}
F)
\begin{alignat*}{4}
x &\,+\,& 2y &\,+\,& z &\,=\,& 3 \\
& & y &\,+\,& 2z &\,=\,& 4 \\
& & y &\,+\,& z &\,=\,& 1
\end{alignat*}
Rank the six linear systems from most complicated to simplest.
Activity 1.2.0.7 .
We can rewrite the previous systems of equations in terms of equivalences of augmented matrices.
\begin{alignat*}{3}
\left[\begin{array}{ccc|c} 2 & 5 & 3 & 7 \\ -1 & -1 & 1 & 1 \\ 1 & 2 & 1 & 3 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 2 & 1 & 3 \\ -1 & -1 & 1 & 1 \\ 2 & 5 & 3 & 7 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 2 & 1 & 3 \\ 0 & 1 & 2 & 4 \\ 2 & 5 & 3 & 7 \end{array}\right] \sim \\
\left[\begin{array}{ccc|c} \circledNumber{1} & 2 & 1 & 3 \\ 0 & \circledNumber{1} & 2 & 4 \\ 0 & 1 & 1 & 1 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 0 & -1 & 1 \\ 0 & \circledNumber{1} & 2 & 4 \\ 0 & 1 & 1 & 1 \end{array}\right] & \sim &
\left[\begin{array}{ccc|c} \circledNumber{1} & 0 & -1 & 1 \\ 0 & \circledNumber{1} & 1 & 1 \\ 0 & 0 & -1 & -3 \end{array}\right]
\end{alignat*}
Determine the row operation(s) necessary in each step to transform the most complicated system's augmented matrix into the simplest.
Activity 1.2.0.8 .
Suppose that the augmented matrix of a system of linear equations has the following shape where \(*\) could be any real number.
\begin{equation*}
\left[
\begin{array}{rrrrr|r}
* \amp * \amp * \amp * \amp * \amp * \\
* \amp * \amp * \amp * \amp * \amp * \\
* \amp * \amp * \amp * \amp * \amp * \\
\end{array}
\right].
\end{equation*}
(a) How many equations are there in this system and how many unknowns?
(b) The most likely solution set is
the empty set (no solution).
exactly one solution.
infinitely many solutions.
(c)
Suppose that this augmented matrix is row equivalent to
\begin{equation*}
\left[
\begin{array}{rrrrr|r}
1 \amp 2 \amp 0 \amp 0 \amp 3 \amp 2 \\
0 \amp 0 \amp 1 \amp 2 \amp -1 \amp -1 \\
0 \amp 0 \amp 0 \amp 0 \amp 0 \amp 0 \\
\end{array}
\right].
\end{equation*}
Make a choice for the names of the unknowns and write the corresponding system of linear equations. Does the system have (A) exactly one solution, (B) infinitely many solutions, or (C) no solutions?
The remaining activities require you to use Gaussian elimination to convert the matrix to reduced row echelon form , abbreviated RREF.
Activity 1.2.0.9 .
Consider the matrix
\begin{equation*}
\left[\begin{array}{cccc}2 & 6 & -1 & 6 \\ 1 & 3 & -1 & 2 \\ -1 & -3 & 2 & 0 \end{array}\right].
\end{equation*}
Which row operation is the best choice for the first move in converting to RREF?
Add row 3 to row 2 (\(R_2+R_3 \rightarrow R_2\) )
Add row 2 to row 3 (\(R_3+R_2 \rightarrow R_3\) )
Swap row 1 to row 2 (\(R_1 \leftrightarrow R_2\) )
Add -2 row 2 to row 1 (\(R_1-2R_2 \rightarrow R_1\) )
Activity 1.2.0.10 .
Consider the matrix
\begin{equation*}
\left[\begin{array}{cccc} \circledNumber{1} & 3 & -1 & 2 \\ 2 & 6 & -1 & 6 \\ -1 & -3 & 2 & 0 \end{array}\right].
\end{equation*}
Which row operation is the best choice for the next move in converting to RREF?
Add row 1 to row 3 (\(R_3+R_1 \rightarrow R_3\) )
Add -2 row 1 to row 2 (\(R_2-2R_1 \rightarrow R_2\) )
Add 2 row 2 to row 3 (\(R_3+2R_2 \rightarrow R_3\) )
Add 2 row 3 to row 2 (\(R_2+2R_3 \rightarrow R_2\) )
Activity 1.2.0.11 .
Consider the matrix
\begin{equation*}
\left[\begin{array}{cccc}\circledNumber{1} & 3 & -1 & 2 \\ 0 & 0 & 1 & 2 \\ 0 & 0 & 1 & 2 \end{array}\right].
\end{equation*}
Which row operation is the best choice for the next move in converting to RREF?
Add row 1 to row 2 (\(R_2+R_1 \rightarrow R_2\) )
Add -1 row 3 to row 2 (\(R_2-R_3 \rightarrow R_2\) )
Add -1 row 2 to row 3 (\(R_3-R_2 \rightarrow R_3\) )
Add row 2 to row 1 (\(R_1+R_2 \rightarrow R_1\) )
Activity 1.2.0.12 .
Use Gaussian elimination to describe the solutions to the following systems of linear equations.
Does the following linear system have (A) exactly one solution, (B) infinitely many solutions, or (C) no solutions?
\begin{equation*}
\begin{alignedat}{4}
x \amp {}+{} \amp y \amp {}+{} \amp 2z \amp {}={} \amp 1 \\
2x \amp {}-{} \amp y \amp {}-{} \amp 2z \amp {}={} \amp 2 \\
-x \amp {}+{} \amp y \amp {}+{} \amp z \amp {}={} \amp 0 \\
\end{alignedat}
\end{equation*}
Does the following linear system have (A) exactly one solution, (B) infinitely many solutions, or (C) no solutions?
\begin{equation*}
\begin{alignedat}{4}
-x \amp {}-{} \amp 2y \amp {}+{} \amp 2z \amp {}={} \amp -1 \\
2x \amp {}+{} \amp 4y \amp {}-{} \amp z \amp {}={} \amp 5 \\
x \amp {}+{} \amp 2y \amp \amp \amp {}={} \amp 3 \\
\end{alignedat}
\end{equation*}
Does the following linear system have (A) exactly one solution, (B) infinitely many solutions, or (C) no solutions?
\begin{equation*}
\begin{alignedat}{4}
-x \amp {}-{} \amp 2y \amp {}+{} \amp 2z \amp {}={} \amp -1 \\
2x \amp {}+{} \amp 4y \amp {}-{} \amp z \amp {}={} \amp 5 \\
x \amp {}+{} \amp 2y \amp \amp \amp {}={} \amp 2 \\
\end{alignedat}
\end{equation*}
The following activities are optional, but provide additional practice with Gaussian elimination.
Activity 1.2.0.13 .
Consider the matrix
\begin{equation*}
\left[\begin{array}{ccc}2 & 1 & 0 \\ 1 & 0 & 0 \\ 3 & -1 & 1 \end{array}\right].
\end{equation*}
(a) Perform three row operations to produce a matrix closer to RREF.
(b) Finish putting it in RREF.
Activity 1.2.0.14 .
Consider the matrix
\begin{equation*}
A=\left[\begin{array}{cccc}2 & 3 & 2 & 3 \\ -2 & 1 & 6 & 1 \\ -1 & -3 & -4 & 1 \end{array}\right].
\end{equation*}
Compute \(\RREF(A)\text{.}\)
Activity 1.2.0.15 .
Consider the matrix
\begin{equation*}
A=\left[\begin{array}{cccc} 2 & 4 & 2 & -4 \\ -2 & -4 & 1 & 1 \\ 3 & 6 & -1 & -4 \end{array}\right].
\end{equation*}
Compute \(\RREF(A)\text{.}\)
