Chapter 8

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{c} 5 x-3 y+2 z=2 \\ 2 x+2 y-3 z=3 \\ -x+7 y-8 z=4 \end{array}\right.$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. $$\left\\{\begin{array}{r} 6 y=42 \\ 6 x-y=16 \end{array}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} y &=e^{x} \\ x-y+1 &=0 \end{aligned}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. $$\left[\begin{array}{rrr} 0 & 2 & -2 \\ 4 & 1 & 2 \end{array}\right]\left(\left[\begin{array}{rr} 4 & 0 \\ 0 & -1 \\ -1 & 2 \end{array}\right]+\left[\begin{array}{rr} -2 & 3 \\ -3 & 5 \\ 0 & -3 \end{array}\right]\right)$$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{7}{x^{2}-14 x}$$

An augmented matrix that represents a system of linear equations (in the variables \(x\) and \(y\) or \(x, y,\) and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[\begin{array}{llll} 1 & 0 & \vdots & -2 \\ 0 & 1 & \vdots & 4 \end{array}\right]$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{l} 2 x+3 y+5 z=4 \\ 3 x+5 y-9 z=7 \\ 5 x+9 y+17 z=13 \end{array}\right.$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. $$\left\\{\begin{aligned} 4 y &=-8 \\ 7 x-2 y &=25 \end{aligned}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} y &=-4 e^{-x} \\ y+3 x+8 &=0 \end{aligned}\right.$$

Operations with Matrices Use the matrix capabilities of a graphing utility to evaluate the expression. $$\left[\begin{array}{r} 3 \\ -1 \\ 5 \\ 7 \end{array}\right]\left(\left[\begin{array}{ll} 5 & -6 \end{array}\right]+\left[\begin{array}{ll} 7 & -1 \end{array}\right]+\left[\begin{array}{ll} -8 & 9 \end{array}\right]\right)$$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x-2}{x^{2}+4 x+3}$$

An augmented matrix that represents a system of linear equations (in the variables \(x\) and \(y\) or \(x, y,\) and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[\begin{array}{lllll} 1 & 0 & 0 & \vdots & -4 \\ 0 & 1 & 0 & \vdots & -8 \\ 0 & 0 & 1 & \vdots & 2 \end{array}\right]$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{rr} 7 x-3 y & +2 w=41 \\ -2 x+y & -w=-13 \\ 4 x+z-2 w & =12 \\ -x+y-x & =-8 \end{array}\right.$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. $$\left\\{\begin{aligned} \frac{3}{2} x-\frac{1}{5} y &=8 \\ -2 x+3 y &=3 \end{aligned}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} x+2 y &=8 \\ y &=2+\ln x \end{aligned}\right.$$

Matrix Multiplication Use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. \(\left\\{\begin{aligned} x+2 y &=4 \\ 3 x+2 y &=0 \end{aligned}\right.\) (a) \(\left[\begin{array}{l}2 \\ 1\end{array}\right]\) (b) \(\left[\begin{array}{r}-2 \\ 3\end{array}\right]\) (c) \(\left[\begin{array}{r}-4 \\ 4\end{array}\right]\) (d) \(\left[\begin{array}{r}2 \\ -3\end{array}\right]\)

An augmented matrix that represents a system of linear equations (in the variables \(x\) and \(y\) or \(x, y,\) and \(z\) ) has been reduced using Gauss-Jordan elimination. Write the solution represented by the augmented matrix. $$\left[\begin{array}{llllr} 1 & 0 & 0 & \vdots & 3 \\ 0 & 1 & 0 & \vdots & -1 \\ 0 & 0 & 1 & \vdots & 0 \end{array}\right]$$

Solve for \(x\) $$\left|\begin{array}{cc} x+1 & 2 \\ -1 & x \end{array}\right|=4$$

Use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. $$\left\\{\begin{array}{c} 2 x+5 y+w=11 \\ x+4 y+2 z-2 w=-7 \\ 2 x-2 y+5 z+w=3 \\ x-3 w=-1 \end{array}\right.$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. $$\left\\{\begin{array}{c} \frac{3}{4} x-\frac{5}{2} y=-9 \\ -x+6 y=28 \end{array}\right.$$

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} 3 y+2 x &=9 \\ y &=-2+\ln (x-1) \end{aligned}\right.$$

Matrix Multiplication Use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. \(\left\\{\begin{aligned} 6 x+2 y &=0 \\\\-x+5 y &=16 \end{aligned}\right.\) (a) \(\left[\begin{array}{r}-1 \\ 3\end{array}\right]\) (b) \(\left[\begin{array}{r}2 \\ -6\end{array}\right]\) (c) \(\left[\begin{array}{r}3 \\ -9\end{array}\right]\) (d) \(\left[\begin{array}{r}-3 \\ 9\end{array}\right]\)

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x^{2}-3 x+2}{4 x^{3}+11 x^{2}}$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{aligned} x+2 y &=7 \\ 2 x+y &=8 \end{aligned}\right.$$

Solve for \(x\) $$\left|\begin{array}{rr} x+3 & 2 \\ 1 & x+2 \end{array}\right|=0$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. \(\left\\{\begin{array}{l}\frac{1}{3} x+y=-\frac{1}{3} \\ 5 x-3 y=7\end{array}\right.\)

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{l} y=\sqrt{x}+4 \\ y=2 x+1 \end{array}\right.$$

Matrix Multiplication Use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. \(\left\\{\begin{aligned}-2 x-3 y &=-6 \\ 4 x+2 y &=20 \end{aligned}\right.\) (a) \(\left[\begin{array}{l}3 \\ 0\end{array}\right]\) (b) \(\left[\begin{array}{l}4 \\ 2\end{array}\right]\) (c) \(\left[\begin{array}{r}-6 \\ 6\end{array}\right]\) (d) \(\left[\begin{array}{r}6 \\ -2\end{array}\right]\)

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{4 x^{2}+3}{(x-5)^{3}}$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{array}{l} 2 x+6 y=16 \\ 2 x+3 y=7 \end{array}\right.$$

Solve for \(x\) $$\left|\begin{array}{rr} x-1 & 2 \\ 3 & x-2 \end{array}\right|=0$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. \(\left\\{\begin{array}{l}5 x-y=-4 \\ 2 x+\frac{3}{5} y=\frac{2}{5}\end{array}\right.\)

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} x-y &=3 \\ \sqrt{x}-y &=1 \end{aligned}\right.$$

Matrix Multiplication Use matrix multiplication to determine whether each matrix is a solution of the system of equations. Use a graphing utility to verify your results. \(\left\\{\begin{array}{l}5 x-7 y=-15 \\ 3 x+y=17\end{array}\right.\) (a) \(\left[\begin{array}{c}-4 \\ -5\end{array}\right]\) (b) \(\left[\begin{array}{l}5 \\ 2\end{array}\right]\) (c) \(\left[\begin{array}{l}4 \\ 5\end{array}\right]\) (d) \(\left[\begin{array}{r}2 \\ 11\end{array}\right]\)

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{6 x+5}{(x+2)^{4}}$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{array}{l} -x+y=-22 \\ 3 x+4 y=4 \\ 4 x-8 y=32 \end{array}\right.$$

Solve for \(x\) $$\left|\begin{array}{cc} 2 x & 1 \\ -1 & x-1 \end{array}\right|=x$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. \(\left\\{\begin{array}{r}0.5 x+2.2 y=9 \\ 6 x+0.4 y=-22\end{array}\right.\)

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{array}{l} x^{2}+y^{2}=169 \\ x^{2}-8 y=104 \end{array}\right.$$

Solving a System of Linear Equations (a) write the system of equations as a matrix equation \(A X=B\) and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix X. Use a graphing utility to check your solution. $$\left\\{\begin{aligned} -x_{1}+x_{2} &=4 \\ -2 x_{1}+x_{2} &=0 \end{aligned}\right.$$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x-1}{x\left(x^{2}+1\right)^{2}}$$

Use a graphing utility to graph the two equations. Use the graphs to approximate the solution of the system. Round your results to three decimal places. \(\left\\{\begin{aligned} 2.4 x+3.8 y &=-17.6 \\ 4 x-0.2 y &=-3.2 \end{aligned}\right.\)

Use a graphing utility to approximate all points of intersection of the graphs of equations in the system. Round your results to three decimal places. Verify your solutions by checking them in the original system. $$\left\\{\begin{aligned} x^{2}+y^{2} &=4 \\ 2 x^{2}-y &=2 \end{aligned}\right.$$

Solving a System of Linear Equations (a) write the system of equations as a matrix equation \(A X=B\) and (b) use Gauss-Jordan elimination on the augmented matrix \([A: B]\) to solve for the matrix X. Use a graphing utility to check your solution. $$\left\\{\begin{array}{c} 2 x_{1}+3 x_{2}=5 \\ x_{1}+4 x_{2}=10 \end{array}\right.$$

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants. $$\frac{x+4}{x^{2}(3 x-1)^{2}}$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{aligned} x+2 y &=0 \\ x+y &=6 \\ 3 x-2 y &=8 \end{aligned}\right.$$

Consider the circuit in the figure. The currents \(I_{1}, I_{2},\) and \(I_{3},\) in amperes, are given by the solution of the system of linear equations \(\left\\{\begin{aligned} 2 I_{1} &+4 I_{3}=E_{1} \\ I_{2}+4 I_{3} &=E_{2} \\\ I_{1}+I_{2}-I_{3} &=0 \end{aligned}\right.\) where \(E_{1}\) and \(E_{2}\) are voltages. Use the inverse of the coefficient matrix of this system to find the unknown currents for the given voltages. \(E_{1}=15\) volts, \(E_{2}=17\) volts

Solve for \(x\) $$\left|\begin{array}{rrr} 1 & 2 & x \\ -1 & 3 & 2 \\ 3 & -2 & 1 \end{array}\right|=0$$

Use any method to solve the system. \(\left\\{\begin{array}{l}3 x-5 y=7 \\ 2 x+y=9\end{array}\right.\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} 2 x-y=0 \\ x^{2}-y=-1 \end{array}\right.$$