Chapter 9

In Problems \(1-26,\) solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{l} 2 x+y=2 \\ 3 x-2 y=-4 \end{array}\right. $$

In Problems 1-8, write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{x-1}{x^{2}+x} $$

In Problems \(1-10,\) determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} x=5 \\ x=y^{2} \end{array}\right. $$

Find the minor and cofactor determinants for each entry in the given determinant. $$ \left|\begin{array}{rr} 4 & 0 \\ 3 & -2 \end{array}\right| $$

In Problems \(1-12\), graph the given inequality. \(x+3 y \geq 6\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{l} 2 x-2 y=1 \\ 3 x+5 y=11 \end{array}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{9 x-8}{x^{2}-1} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} y=3 \\ (x+1)^{2}+y^{2}=10 \end{array}\right. $$

Find the minor and cofactor determinants for each entry in the given determinant. $$ \left|\begin{array}{rr} 6 & -2 \\ 5 & 1 \end{array}\right| $$

Graph the given inequality. \(x-y \leq 4\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{r} 4 x-y+1=0 \\ x+3 y+9=0 \end{array}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{2 x^{2}-3}{x^{3}+x^{2}} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{aligned} -x^{2}+y &=-1 \\ x^{2}+y &=4 \end{aligned}\right. $$

Find the minor and cofactor determinants for each entry in the given determinant. $$ \left|\begin{array}{rrr} 1 & -7 & 8 \\ 2 & 1 & 0 \\ -3 & 0 & 5 \end{array}\right| $$

Graph the given inequality. \(x+2 y<-x+3 y\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{r} x-4 y+1=0 \\ 3 x+2 y-1=0 \end{array}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{2 x^{2}-3}{x^{3}+x^{2}} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} x+y=5 \\ x^{2}+y^{2}=1 \end{array}\right. $$

Find the minor and cofactor determinants for each entry in the given determinant. $$ \left|\begin{array}{rrr} 4 & -3 & 0 \\ 2 & -1 & 6 \\ -5 & 4 & 1 \end{array}\right| $$

Graph the given inequality. \(2 x+5 y>x-y+6\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{r} x-2 y=6 \\ -0.5 x+y=1 \end{array}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{3 x^{2}-x+4}{x^{4}+2 x^{3}+x^{2}} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=1 \\ x^{2}-4 x+y^{2}=-3 \end{array}\right. $$

Evaluate the given determinant. In Problem 10 , assume that \(a \neq 0, b \neq 0\). $$ \left|\begin{array}{cc} \frac{5}{3} & \frac{1}{2} \\ 6 & 18 \end{array}\right| $$

Graph the given inequality. \(-y \geq 2(x+3)-5\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{aligned} 6 x-4 y &=9 \\ -3 x+2 y &=-4.5 \end{aligned}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{4}{x^{3}\left(x^{2}+1\right)} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} y=2^{x}-1 \\ y=\log _{2}(x+2) \end{array}\right. $$

Evaluate the given determinant. In Problem 10 , assume that \(a \neq 0, b \neq 0\). $$ \left|\begin{array}{rr} 0 & -1 \\ 8 & 0 \end{array}\right| $$

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{l} x-y=2 \\ x+y=1 \end{array}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{2 x^{3}-x}{\left(x^{2}+1\right)^{2}} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} y-x^{2}=0 \\ x^{2}-y^{2}=4 \end{array}\right. $$

Evaluate the given determinant. In Problem 10 , assume that \(a \neq 0, b \neq 0\). $$ \left|\begin{array}{ll} 4 & 2 \\ 0 & 3 \end{array}\right| $$

Graph the given inequality. \(y \geq(x-1)^{2}\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{l} 2 x+y=4 \\ 2 x+y=0 \end{array}\right. $$

Write out the appropriate form of the partial fraction decomposition of the given rational expression. Do not evaluate the coefficients. $$ \frac{-x^{2}+3 x+7}{\left(x^{2}+x-2\right)\left(x^{2}+x+1\right)^{3}} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} y=-x^{2}+2 x \\ (x-1)^{2}+y^{2}=1 \end{array}\right. $$

Evaluate the given determinant. In Problem 10 , assume that \(a \neq 0, b \neq 0\). $$ \left|\begin{array}{rr} 3 & -4 \\ 5 & 6 \end{array}\right| $$

Graph the given inequality. \(x^{2}+\frac{1}{4} y^{2}<1\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{r} -x-2 y+4=0 \\ 5 x+10 y-20=0 \end{array}\right. $$

In Problems 9-32, find the partial fraction decomposition of the given rational expression. $$ \frac{1}{x(x+2)} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} y=\sqrt{x} \\ y=2^{-x} \end{array}\right. $$

Evaluate the given determinant. In Problem 10 , assume that \(a \neq 0, b \neq 0\). $$ \left|\begin{array}{rr} a & -b \\ b & a \end{array}\right| $$

Graph the given inequality. \(y-1 \leq \sqrt{x}\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{r} 7 x-3 y-14=0 \\ x+y-1=0 \end{array}\right. $$

Find the partial fraction decomposition of the given rational expression. $$ \frac{2}{x(4 x-1)} $$

Determine graphically whether the given nonlinear system has any real solutions. $$ \left\\{\begin{array}{l} x^{2}+y^{2}=5 \\ (x-y)^{2}=1 \end{array}\right. $$

Evaluate the given determinant. In Problem 10 , assume that \(a \neq 0, b \neq 0\). $$ \left|\begin{array}{ll} a & b \\ \frac{1}{b} & \frac{1}{a} \end{array}\right| $$

Graph the given inequality. \(y \geq \sqrt{x+1}\)

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{c} x+y-z=0 \\ x-y+z=2 \\ 2 x+y-4 z=-8 \end{array}\right. $$