Chapter 6

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rrr|r} 1 & 1 & -1 & 4 \\ 0 & 1 & -1 & 2 \\ 0 & 0 & 1 & 1 \end{array}\right] $$

If possible, solve the system. $$ \begin{aligned} &\begin{array}{l} x-4 y+2 z=-2 \\ x+2 y-2 z=-3 \end{array}\\\ &x-y \quad \quad \quad=4 \end{aligned} $$

Solve the equation for \(x\) and then solve it for \(y .\) $$ \frac{2 x-y}{3 y}=1 $$

If possible, find each of the following. (a) \(A+B\) (b) \(3 A\) (c) \(2 A-3 B\) $$A=\left[\begin{array}{rrr}6 & 2 & 9 \\\3 & -2 & 0 \\\\-1 & 4 & 8\end{array}\right]$$ $$B=\left[\begin{array}{rrr}1 & 0 & -1 \\\3 & 0 & 7 \\\0 & -2 & -5\end{array}\right]$$

Graph the solution set to the system of inequalities. Use the graph to identify one solution. $$ \begin{aligned} &y \leq \sqrt{x}\\\ &y \geq 1 \end{aligned} $$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{rrr} 3 & 0 & -1 \\ 2 & 3 & -4 \\ 6 & -5 & 1 \end{array}\right] $$

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{ll} 1 & 3 \\ 2 & 5 \end{array}\right] $$

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rrr|r} 1 & -2 & -1 & 0 \\ 0 & 1 & -3 & 1 \\ 0 & 0 & 1 & 2 \end{array}\right] $$

If possible, solve the system. $$ \begin{array}{r} 2 x+y+3 z=4 \\ -3 x-y-4 z=5 \\ x+y+2 z=0 \end{array} $$

Solve the equation for \(x\) and then solve it for \(y .\) $$ \frac{x+y}{x-y}=2 $$

If possible, find each of the following. (a) \(A+B\) (b) \(3 A\) (c) \(2 A-3 B\) $$A=\left[\begin{array}{rr}-2 & -1 \\\\-5 & 1 \\\2 & -3\end{array}\right]$$ $$B=\left[\begin{array}{rr}2 & -1 \\\3 & 1 \\\7 & -5\end{array}\right]$$

Graph the solution set to the system of inequalities. Use the graph to identify one solution. $$ \begin{array}{l} x+2 y>-2 \\ x+2 y<5 \end{array} $$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{rrr} 1 & -5 & 2 \\ -7 & 1 & 3 \\ 0 & 4 & -2 \end{array}\right] $$

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rrr|r} 1 & 2 & -1 & 5 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

Determine which ordered pairs are solutions to the given system of equations. State whether the system is linear or nonlinear. $$ \begin{array}{l} (2,1),(-2,1),(1,0) \\ 2 x+y=5 \\ x+y=3 \end{array} $$

If possible, solve the system. $$ \begin{aligned} &4 a-b+2 c=0\\\ &\begin{array}{l} 2 a+b-c=-11 \\ 2 a-2 b+c=3 \end{array} \end{aligned} $$

Graph the solution set to the system of inequalities. Use the graph to identify one solution. $$ \begin{array}{l} x-y \leq 3 \\ x+y \leq 3 \end{array} $$

Let \(A\) be the given matrix. Find det \(A\) by using the method of co factors. $$ \left[\begin{array}{rrr} 1 & -1 & 2 \\ -2 & 0 & 1 \\ 1 & 1 & -1 \end{array}\right] $$

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{ll} -2 & 4 \\ -5 & 9 \end{array}\right] $$

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rrr|r} 1 & -1 & 2 & 8 \\ 0 & 1 & -4 & 2 \\ 0 & 0 & 0 & 0 \end{array}\right] $$

Determine which ordered pairs are solutions to the given system of equations. State whether the system is linear or nonlinear. $$ \begin{aligned} &\begin{array}{l} (3,2),(3,-4),(5,0) \\ x-y=5 \end{array}\\\ &2 x+y=10 \end{aligned} $$

If possible, solve the system. $$ \begin{array}{r} a-4 b+3 c=2 \\ -a-2 b+5 c=9 \\ a+2 b+c=6 \end{array} $$

Evaluate the matrix expression. $$2\left[\begin{array}{rr}2 & -1 \\\5 & 1 \\\0 & 3\end{array}\right]+\left[\begin{array}{rr} 5 & 0 \\\7 & -3 \\\1 & 1\end{array}\right]-\left[\begin{array}{rr}9 & -4 \\\4 & 4 \\\1 & 6 \end{array}\right]$$

Graph the solution set to the system of inequalities. Use the graph to identify one solution. $$ \begin{aligned} &x^{2}+y^{2} \leq 16\\\ &x+y<2 \end{aligned} $$

Let \(A\) be the given matrix. Use technology to calculate det \(A\). $$ \left[\begin{array}{rr} 11 & -32 \\ 1.2 & 55 \end{array}\right] $$

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{lll} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{array}\right] $$

Determine which ordered pairs are solutions to the given system of equations. State whether the system is linear or nonlinear. $$ \begin{aligned} &(4,-3),(0,5),(4,3)\\\ &x^{2}+y^{2}=25\\\ &2 x+3 y=-1 \end{aligned} $$

If possible, solve the system. $$ \begin{array}{l} a+b+c=0 \\ a-b-c=3 \\ a+3 b+3 c=5 \end{array} $$

Evaluate the matrix expression. $$-3\left[\begin{array}{rr}3 & 8 \\\\-1 & -9\end{array}\right]+5\left[\begin{array}{rr}4 & -8 \\\1 & 6\end{array}\right]$$

Graph the solution set to the system of inequalities. Use the graph to identify one solution. $$ \begin{array}{l} x^{2}+y \leq 4 \\ x^{2}-y \leq 3 \end{array} $$

Let \(A\) be the given matrix. Use technology to calculate det \(A\). $$ \left[\begin{array}{rrr} 17 & -4 & 3 \\ 11 & 5 & -15 \\ 7 & -9 & 23 \end{array}\right] $$

The augmented matrix is in row-echelon form and represents a linear system. Solve the system by using backwand substitution, if possible. Write the solution as either an ordered pair or an ordered triple. $$ \left[\begin{array}{rrr|r} 1 & 0 & -4 & \frac{3}{4} \\ 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & -3 \end{array}\right] $$

Determine which ordered pairs are solutions to the given system of equations. State whether the system is linear or nonlinear. $$ \begin{array}{l} (4,8),(8,4),(-4,-8) \\ \quad x y=32 \\ x+y=12 \end{array} $$

If possible, solve the system. $$ \begin{aligned} a-2 b+c &=-1 \\ a+5 b &=-3 \\ 2 a+3 b+c &=-2 \end{aligned} $$

Evaluate the matrix expression. $$\left[\begin{array}{rr}4 & 6 \\\3 & -7\end{array}\right]-2\left[\begin{array}{rr}1 & 0 \\\\-4 & 1\end{array}\right]$$

Graph the solution set to the system of inequalities. $$ \begin{array}{c} x+2 y \leq 4 \\ 2 x-y \geq 6 \end{array} $$

Let \(A\) be the given matrix. Use technology to calculate det \(A\). $$ \left[\begin{array}{rrr} 2.3 & 5.1 & 2.8 \\ 1.2 & 4.5 & 8.8 \\ -0.4 & -0.8 & -1.2 \end{array}\right] $$

If possible, solve the system. $$ \begin{array}{rr} 3 x+2 y+z= & -1 \\ 3 x+4 y-z= & 1 \\ x+2 y+z= & 0 \end{array} $$

Evaluate the matrix expression. $$\left[\begin{array}{rrr}5 & -1 & 6 \\\\-2 & 10 & 12 \\\5 & 2 & 9\end{array}\right]-\left[\begin{array}{rrr}-1 & 2 & 2 \\\2 & -1 & 2 \\\2 & 2 & -1\end{array}\right]$$

Graph the solution set to the system of inequalities. $$ \begin{array}{r} 3 x-y \leq 3 \\ x+2 y \leq 2 \end{array} $$

( Refer to Examples 3-5.) LetA be the given matrix. Find \(A^{-1}\) without a calculator. $$ \left[\begin{array}{rrr} -2 & 1 & 0 \\ 1 & 0 & 1 \\ -1 & 1 & 0 \end{array}\right] $$

Let \(A\) be the given matrix. Use technology to calculate det \(A\). $$ \left[\begin{array}{rrrr} 1 & -1 & 3 & 7 \\ 9 & 2 & -7 & -4 \\ 5 & -7 & 1 & -9 \\ 7 & 1 & 3 & 6 \end{array}\right] $$

If possible, solve the system. $$ \begin{aligned} x-2 y+z &=1 \\ x+y+2 z &=2 \\ 2 x+3 y+z &=6 \end{aligned} $$

Evaluate the matrix expression. $$2\left[\begin{array}{rrr}2 & -1 & -1 \\\\-1 & 2 & -1 \\\\-1 & -1 & 2\end{array}\right]+3\left[\begin{array}{lll}1 & 2 & 3 \\\2 & 1 & 3 \\\2 & 3 & 1\end{array}\right]$$

Graph the solution set to the system of inequalities. $$ \begin{array}{r} 3 x+2 y<6 \\ x+3 y \leq 6 \end{array} $$

Use Cramer's rule to solve the system of linear equations. $$ \begin{array}{l} -x+2 y=5 \\ 3 x+3 y=1 \end{array} $$

Perform each row operation on the given matrix by completing the matrix at the right. $$ \left[\begin{array}{rrr|r} 1 & -1 & 1 & 2 \\ -1 & 2 & -2 & 0 \\ 1 & 7 & 0 & 5 \end{array}\right] \begin{array}{l} R_{2}+R_{1} \rightarrow \\ R_{3}-R_{1} \rightarrow \end{array}\left[\begin{array}{rrr|r} 1 & -1 & 1 & 2 \\ & & & \end{array}\right] $$

If possible, solve the system. $$ \begin{array}{l} -x+3 y+z=3 \\ 2 x+7 y+4 z=13 \\ 4 x+y+2 z=7 \end{array} $$

Evaluate the matrix expression. $$3\left[\begin{array}{rrrr}1 & 0 & 3 & -1 \\\0 & 1 & 2 & -1 \\\1 & 0 & -3 & 1\end{array}\right]-4\left[\begin{array}{rrrr}-1 & 0 & 0 & 4 \\\0 & -1 & 3 & 2 \\\2 & 0 & 1 & -1\end{array}\right]$$

Graph the solution set to the system of inequalities. $$ \begin{aligned} &4 x+3 y \geq 12\\\ &2 x+6 y \geq 4 \end{aligned} $$