Chapter 11

Subscript notation is frequently used for working with larger systems of equations. For Problems 31-34, use a matrix approach to solve each system. Express the solutions as 4-tuples of the form \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\).\(\left(\begin{array}{rl}x_{1}+2 x_{2}-3 x_{3}+x_{4} & =-2 \\ -2 x_{1}-3 x_{2}+x_{3}-x_{4} & =5 \\ 4 x_{1}+9 x_{2}-2 x_{3}-2 x_{4} & =-28 \\\ -5 x_{1}-9 x_{2}+2 x_{3}-3 x_{4} & =14\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}5 x-3 y=-34 \\ 2 x+7 y=-30\end{array}\right)\)

A linear system in which the constant terms are all zero is called a homogeneous system. (a) Verify that for a \(3 \times 3\) homogeneous system, if \(D \neq\) 0 , then \((0,0,0)\) is the only solution for the system. (b) Verify that for a \(3 \times 3\) homogeneous system, if \(D=\) 0 , then the equations are dependent.

Use the appropriate property of determinants from this section to justify each true statement. Do not evaluate the determinants. \(\left|\begin{array}{rrr}4 & 7 & 9 \\ 6 & -8 & 2 \\ 4 & 3 & -1\end{array}\right|=-\left|\begin{array}{rrr}4 & 9 & 7 \\ 6 & 2 & -8 \\ 4 & -1 & 3\end{array}\right|\)

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{llll:r}1 & 0 & 0 & 0 & -2 \\ 0 & 1 & 0 & 0 & 4 \\ 0 & 0 & 1 & 0 & -3 \\ 0 & 0 & 0 & 1 & 0\end{array}\right]\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}5 x-2 y=19 \\ 5 x-2 y=7\end{array}\right)\)

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{rrrr:r}1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & -5 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 4\end{array}\right]\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}4 a+2 b=-4 \\ 6 a-5 b=18\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{r}2 x-y+z=0 \\ 3 x+2 y+5 z=0 \\ 4 x-7 y+z=0\end{array}\right)\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}5 a+6 b=8 \\ 2 a-15 b=9\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{r}3 x+y-z=0 \\ x-y+2 z=0 \\ 4 x-5 y-2 z=0\end{array}\right)\)

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{rrrr:r}1 & 0 & 0 & 0 & 2 \\ 0 & 1 & 0 & 2 & -3 \\ 0 & 0 & 1 & 3 & 4 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}7 x+2 y=11 \\ 7 x+2 y=-4\end{array}\right)\)

Use Cramer's rule to find the solution set for each system. If the equations are dependent, simply indicate that there are infinitely many solutions. \(\left(\begin{array}{r}2 x-y+2 z=0 \\ x+2 y+z=0 \\ x-3 y+z=0\end{array}\right)\)

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{rrrr:r}1 & 0 & 0 & 3 & 5 \\ 0 & 1 & 0 & 0 & -1 \\ 0 & 0 & 1 & 4 & 2 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}\frac{2}{3} s+\frac{1}{4} t=-1 \\ \frac{1}{2} s-\frac{1}{3} t=-7\end{array}\right)\)

Use the appropriate property of determinants from this section to justify each true statement. Do not evaluate the determinants. \(\left|\begin{array}{rrr}2 & 1 & -3 \\ 0 & 2 & -4 \\ -5 & 1 & 3\end{array}\right|=-\left|\begin{array}{rrr}2 & 1 & -3 \\ -5 & 1 & 3 \\ 0 & 2 & -4\end{array}\right|\)

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{llll:l}1 & 3 & 0 & 2 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}\frac{1}{4} s-\frac{2}{3} t=-3 \\ \frac{1}{3} s+\frac{1}{3} t=7\end{array}\right)\)

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{rrrr:r}1 & 3 & 0 & 0 & 9 \\ 0 & 0 & 1 & 0 & 2 \\ 0 & 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Solve each system by using the substitution method. \(\left(\begin{array}{l}\frac{x}{2}-\frac{2 y}{5}=\frac{-23}{60} \\ \frac{2 x}{3}+\frac{y}{4}=\frac{-1}{4}\end{array}\right)\)

In Problems 35-42, each matrix is the reduced echelon matrix for a system with variables \(x_{1}, x_{2}, x_{3}\), and \(x_{4}\). Find the solution set of each system. \(\left[\begin{array}{rrrr:r}1 & 0 & 0 & 0 & 7 \\ 0 & 1 & 0 & 0 & -3 \\ 0 & 0 & 1 & -2 & 5 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\)

Solve each system by using the substitution method. \(\left(\begin{array}{c}\frac{2 x}{3}-\frac{y}{2}=\frac{3}{5} \\\ \frac{x}{4}+\frac{y}{2}=\frac{7}{80}\end{array}\right)\)

Explain the difference between a matrix and a determinant.

What is a matrix? What is an augmented matrix of a system of linear equations?

Solve each system by using the substitution method. \(\left(\begin{array}{rl}\frac{2}{3} x+\frac{1}{2} y & =\frac{1}{6} \\ 4 x+6 y & =-1\end{array}\right)\)

Explain the concept of a cofactor and how it is used to help expand a determinant.

Describe how to use matrices to solve the system $$ \left(\begin{array}{r} x-2 y=5 \\ 2 x+7 y=9 \end{array}\right) \text {. } $$

Solve each system by using the substitution method. \(\left(\begin{array}{rl}\frac{1}{2} x+\frac{2}{3} y & =-\frac{3}{10} \\ 5 x+4 y & =-1\end{array}\right)\)

What does it mean to say that any row or column can be used to expand a determinant?

For Problems \(45-50\), change each augmented matrix of the system to reduced echelon form and then indicate the solutions of the system. \(\left(\begin{array}{l}x-2 y+3 z=4 \\ 3 x-5 y-z=7\end{array}\right)\)

For Problems \(45-60\), solve each system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. \(\left(\begin{array}{rl}5 x-y & =-22 \\ 2 x+3 y & =-2\end{array}\right)\)

Give a step-by-step explanation of how to evaluate the determinant $$ \left|\begin{array}{rrr} 3 & 0 & 2 \\ 1 & -2 & 5 \\ 6 & 0 & 9 \end{array}\right| $$

For Problems \(45-50\), change each augmented matrix of the system to reduced echelon form and then indicate the solutions of the system. \(\left(\begin{array}{c}x+3 y-2 z=-1 \\ 2 x-5 y+7 z=4\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}4 x+5 y=-41 \\ 3 x-2 y=21\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}x=3 y-10 \\ x=-2 y+15\end{array}\right)\)

For Problems \(45-50\), change each augmented matrix of the system to reduced echelon form and then indicate the solutions of the system. \(\left(\begin{array}{r}3 x+6 y-z=9 \\ 2 x-3 y+4 z=1\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}y=4 x-24 \\ 7 x+y=42\end{array}\right)\)

For Problems \(45-50\), change each augmented matrix of the system to reduced echelon form and then indicate the solutions of the system. \(\left(\begin{array}{r}x-2 y+4 z=9 \\ 2 x-4 y+8 z=3\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{c}y=\frac{2}{5} x-3 \\ 4 x-7 y=33\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}\frac{1}{2} x-\frac{2}{3} y=22 \\ \frac{1}{2} x+\frac{1}{4} y=0\end{array}\right)\)

Consider the following matrix: $$ A=\left[\begin{array}{rrrr} 2 & 5 & 7 & 9 \\ -4 & 6 & 2 & 4 \\ 6 & 9 & 12 & 3 \\ 5 & 4 & -2 & 8 \end{array}\right] $$ Form matrix \(B\) by interchanging rows 1 and 3 of ma\(\operatorname{trix} A\). Now use your calculator to show that \(|B|=-|A|\).

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left.\begin{array}{l}\frac{2}{5} x-\frac{1}{3} y=-9 \\ \frac{3}{4} x+\frac{1}{3} y=-14\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}t=2 u+2 \\ 9 u-9 t=-45\end{array}\right)\)

Consider the following matrix: $$ A=\left[\begin{array}{rrrrrr} 4 & 3 & 2 & 1 & 5 & -3 \\ 5 & 2 & 7 & 8 & 6 & 3 \\ 0 & 9 & 1 & 4 & 7 & 2 \\ 4 & 3 & 2 & 1 & 5 & -3 \\ -4 & -6 & 7 & 12 & 11 & 9 \\ 5 & 8 & 6 & -3 & 2 & -1 \end{array}\right] $$ Use your calculator to show that \(|A|=0\).

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}9 u-9 t=36 \\ u=2 t+1\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left.\begin{array}{rl}x+y & =1000 \\ 0.12 x+0.14 y & =136\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{rl}x+y & =10 \\ 0.3 x+0.7 y & =4\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left.\begin{array}{rl}y & =2 x \\ 0.09 x+0.12 y & =132\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{rl}y & =3 x \\ 0.1 x+0.11 y & =64.5\end{array}\right)\)