Finding solutions to linear systems

For each of the following linear systems, use Gaussian elimination to describe the solutions to the following systems of linear equations. In particular, determine whether each linear system has exactly one solution, infinitely many solutions, or no solutions.

1. $\begin{alignedat}{4} x & {}+{} & y & {}+{} & 2z & {}={} & 1 \\ 2x & {}-{} & y & {}-{} & 2z & {}={} & 2 \\ -x & {}+{} & y & {}+{} & z & {}={} & 0 \\ \\ \end{alignedat}$

2. $\begin{alignedat}{4} -x & {}-{} & 2y & {}+{} & 2z & {}={} & -1 \\ 2x & {}+{} & 4y & {}-{} & z & {}={} & 5 \\ x & {}+{} & 2y & & & {}={} & 3 \\ \\ \end{alignedat}$

3. $\begin{alignedat}{4} -x & {}-{} & 2y & {}+{} & 2z & {}={} & -1 \\ 2x & {}+{} & 4y & {}-{} & z & {}={} & 5 \\ x & {}+{} & 2y & & & {}={} & 2 \\ \\ \end{alignedat}$

1. Write the augmented matrix for the linear system and perform Gaussian elimination to describe the solution space in as much detail as you can.

2. Suppose that you have a linear system in the variables $x$ and $y$ whose augmented matrix is row equivalent to Write the linear system corresponding to this augmented matrix and describe its solution set in as much detail as you can.

3. Suppose that you have a lin…

Consider each of the following augmented matrices. Determine if the matrix is in reduced row echelon form. If it is not, perform a sequence of scaling, interchange, and replacement operations to obtain a row equivalent matrix that is in reduced row echelon form. Then use the reduced row echelon matrix to describe the solution space.

1. $\left[ \begin{array}{rrr|r} 2 & 0 & 4 & -8 \\ 0 & 1 & 3 & 2 \\ \end{array} \right].$

2. $\left[ \begin{array}{rrr|r} 1 & 0 & 0 & -1 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & 1 \\ \end{array} \right].$

3. $\left[ \begin{array}{rrr|r} 1 & 0 & 4 & 2 \\ 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 1 \\ \end{array} \right].$

4. $\left[ \begin{array}{rrr|r} 0 & 1 & 3 & 2 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 4 & 2 \\ \end{array} \right].$

5. $\left[ \begin{array}{rrr|r} 1 & 2 & -1 & 2 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 1 & 1 \\ \end{array} \right].$

For each of the linear systems below, write the associated augmented matrix and find the reduced row echelon matrix that is row equivalent to it. Identify the basic and free variables and then describe the solution space of the original linear system using a parametric description, if appropriate.

Consider each matrix below and determine if it is in reduced row echelon form. If not, indicate the reason and apply a sequence of row operations to find its reduced row echelon matrix. For each matrix, indicate whether the corresponding linear system has infinitely many solutions, exactly one solution, or no solutions.

Give an example of a reduced row echelon matrix that describes a linear system having the stated properties. If it is not possible to find such an example, explain why not.

1. Write a reduced row echelon matrix for a linear system having five equations and three variables and having exactly one solution.

2. Write a reduced row echelon matrix for a linear system having three equations and three variables a…

For each of the questions below, provide a justification for your response.

1. What does the presence of a row whose entries are all zero in an augmented matrix tell us about the solution space of the linear system?

2. How can you determine if a linear system has no solutions directly from its reduced row echelon matrix?

3. How can you determine if a linear system has infinitely many solutions directly from i…

Determine whether the following statements are true or false and explain your reasoning.

1. If every variable is basic, then the linear system has exactly one solution.

2. If two augmented matrices are row equivalent to one another, then they describe two linear systems having the same solution spaces.

3. The presence of a free variable indicates that there are no solutions to the linear system.

4. If a linear sy…