Chapter 8

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. $$\left\\{\begin{aligned} -2 x_{1}-3 x_{2} &=-4 \\ 6 x_{1}+x_{2} &=-36 \end{aligned}\right.$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{1}{x^{2}+x}$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{aligned} x+2 y-3 z &=-28 \\ 4 y+2 z &=0 \\ -x+y-z &=-5 \end{aligned}\right.$$

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. \(E_{1}=10\) volts, \(E_{2}=10\) volts

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

Use any method to solve the system. \(\left\\{\begin{array}{r}-x+3 y=17 \\ 4 x+3 y=7\end{array}\right.\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{r} x+y=4 \\ x^{2}+y=2 \end{array}\right.$$

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. $$\left\\{\begin{aligned} -4 x_{1}+9 x_{2} &=-13 \\ x_{1}-3 x_{2} &=12 \end{aligned}\right.$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{1}{4 x^{2}-9}$$

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 a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials. 500 units of sand 500 units of loam 400 units of peat moss

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} 4 u & -1 \\ -1 & 2 v \end{array}\right|$$

Use any method to solve the system. \(\left\\{\begin{array}{l}y=2 x-5 \\ y=5 x-11\end{array}\right.\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} 3 x-7 y=-6 \\ x^{2}-y^{2}=4 \end{array}\right.$$

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. $$\left\\{\begin{array}{rr} x_{1}-2 x_{2}+3 x_{3}= & 9 \\ -x_{1}+3 x_{2}-x_{3}= & -6 \\ 2 x_{1}-5 x_{2}+5 x_{3}= & 17 \end{array}\right.$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{5-x}{2 x^{2}+x-1}$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{aligned} 3 x+2 y-z+w &=0 \\ x-y+4 z+2 w &=25 \\ -2 x+y+2 z-w &=2 \\ x+y+z+w &=6 \end{aligned}\right.$$

Consider a company that specializes in potting soil. Each bag of potting soil for seedlings requires 2 units of sand, 1 unit of loam, and 1 unit of peat moss. Each bag of potting soil for general potting requires 1 unit of sand, 2 units of loam, and 1 unit of peat moss. Each bag of potting soil for hardwood plants requires 2 units of sand, 2 units of loam, and 2 units of peat moss. Find the numbers of bags of the three types of potting soil that the company can produce with the given amounts of raw materials. 500 units of sand 750 units of loam 450 units of peat moss

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} 3 x^{2} & -3 y^{2} \\ 1 & 1 \end{array}\right|$$

Use any method to solve the system. \(\left\\{\begin{aligned} 7 x+3 y &=16 \\ y &=x+2 \end{aligned}\right.\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} x^{2}+y^{2}=25 \\ 2 x+y=10 \end{array}\right.$$

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. $$\left\\{\begin{array}{rr} x_{1}+x_{2}-3 x_{3}= & 9 \\ -x_{1}+2 x_{2} & =6 \\ x_{1}-x_{2}+x_{3} & =-5 \end{array}\right.$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x-2}{x^{2}+4 x+3}$$

Use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution. $$\left\\{\begin{array}{rr} x-4 y+3 z-2 w= & 9 \\ 3 x-2 y+z-4 w= & -13 \\ -4 x+3 y-2 z+w= & -4 \\ -2 x+y-4 z+3 w= & -10 \end{array}\right.$$

A florist is creating 10 centerpieces for the tables at a wedding reception. Roses cost \(\$ 2.50\) each, lilies cost \(\$ 4\) each, and irises cost \(\$ 2\) each. The customer has a budget of \(\$ 300\) for the centerpieces and wants each centerpiece to contain 12 flowers, with twice as many roses as the number of irises and lilies combined. (a) Write a linear system that represents the situation. (b) Write a matrix equation that corresponds to your system. (c) Solve your linear system using an inverse matrix. Find the number of flowers of each type that the florist can use to create the 10 centerpieces.

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} e^{2 x} & e^{3 x} \\ 2 e^{2 x} & 3 e^{3 x} \end{array}\right|$$

Use any method to solve the system. \(\left\\{\begin{array}{c}x-5 y=21 \\ 6 x+5 y=21\end{array}\right.\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{c} x^{2}+y^{2}=1 \\ x+y=4 \end{array}\right.$$

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. $$\left\\{\begin{aligned} x_{1}-5 x_{2}+2 x_{3} &=-20 \\ -3 x_{1}+x_{2}-x_{3} &=8 \\ -2 x_{2}+5 x_{3} &=-16 \end{aligned}\right.$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x^{2}+12 x+12}{x^{3}-4 x}$$

Use matrices to solve the system of equations, if possible. Use Gauss-Jordan elimination. $$\left\\{\begin{array}{c} x-3 z=-2 \\ 3 x+y-2 z=5 \\ 2 x+2 y+z=4 \end{array}\right.$$

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} e^{-x} & x e^{-x} \\ -e^{-x} & (1-x) e^{-x} \end{array}\right|$$

Use any method to solve the system. \(\left\\{\begin{array}{l}y=-2 x-17 \\ y=2-3 x\end{array}\right.\)

Solve the system graphically or algebraically. Explain your choice of method. $$\left\\{\begin{array}{l} x^{2}+y^{2}=4 \\ x-y=5 \end{array}\right.$$

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. $$\left\\{\begin{array}{rr} x_{1}-x_{2}+4 x_{3}= & 17 \\ x_{1}+3 x_{2} & =-11 \\ -6 x_{2}+5 x_{3} & =40 \end{array}\right.$$

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x^{2}+12 x-9}{x^{3}-9 x}$$

Use matrices to solve the system of equations, if possible. Use Gauss-Jordan elimination. $$\left\\{\begin{aligned} 2 x-y+3 z &=24 \\ 2 y-z &=14 \\ 7 x-5 y &=6 \end{aligned}\right.$$

Determine whether the statement is true or false. Justify your answer. Multiplication of an invertible matrix and its inverse is commutative.

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{ll} x & \ln x \\ 1 & 1 / x \end{array}\right|$$

Use any method to solve the system. \(\left\\{\begin{aligned}-5 x+9 y &=13 \\ y &=x-4 \end{aligned}\right.\)

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

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$. (a) \(A(B+C)\) (b) \(A B+A C\)

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{4 x^{2}+2 x-1}{x^{2}(x+1)}$$

Use matrices to solve the system of equations, if possible. Use Gauss-Jordan elimination. $$\left\\{\begin{array}{rr} -x-y-3 z= & -12 \\ 2 x-y-4 z= & 6 \\ -2 x+4 y+14 z= & 19 \end{array}\right.$$

Determine whether the statement is true or false. Justify your answer. When the product of two square matrices is the identity matrix, the matrices are inverses of one another.

Evaluate the determinant, in which the entries are functions. Determinants of this type occur when changes of variables are made in calculus. $$\left|\begin{array}{cc} x & x \ln x \\ 1 & 1+\ln x \end{array}\right|$$

Use any method to solve the system. \(\left\\{\begin{array}{r}4 x-3 y=6 \\ -5 x+7 y=-1\end{array}\right.\)

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

Operations with Matrices Use a graphing utility to perform the operations for the matrices \(A, B,\) and \(C\) and the scalar \(c .\) Write a brief statement comparing the results of parts (a) and (b). $$\begin{aligned} &A=\left[\begin{array}{rrr} 1 & 2 & -1 \\ 0 & -2 & 3 \\ 4 & -3 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 2 & 3 & 0 \\ 4 & 1 & -2 \\ -1 & 2 & 0 \end{array}\right]\\\ &C=\left[\begin{array}{rrr} 3 & -2 & 1 \\ -4 & 0 & 3 \\ -1 & 3 & -2 \end{array}\right], \text { and } c=3 \end{aligned}$$ (a) \((B+C) A\) (b) \(B A+C A\)

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{2 x-3}{(x-1)^{2}}$$