Systems of Equations and Inequalities

In Problems 13–16, use the following matrices to compute each expression.

$A=\left[\begin{array}{cc}1& 0\\ 2& 4\\ -1& 2\end{array}\right] B=\left[\begin{array}{ccc}4& -3& 0\\ 1& 1& -2\end{array}\right] C=\left[\begin{array}{cc}3& -4\\ 1& 5\\ 5& 2\end{array}\right]$

$AB$

In Problems 13–16, use the following matrices to compute each expression.

$A=\left[\begin{array}{cc}1& 0\\ 2& 4\\ -1& 2\end{array}\right] B=\left[\begin{array}{ccc}4& -3& 0\\ 1& 1& -2\end{array}\right] C=\left[\begin{array}{cc}3& -4\\ 1& 5\\ 5& 2\end{array}\right]$

$BC$

In Problems 17–19, find the inverse, if there is one, of each matrix. If there is not an inverse, say that the matrix is singular.

$\left[\begin{array}{cc}4& 6\\ 1& 3\end{array}\right]$

In Problems 17–19, find the inverse, if there is one, of each matrix. If there is not an inverse, say that the matrix is singular.

$\left[\begin{array}{ccc}1& 3& 3\\ 1& 2& 1\\ 1& -1& 2\end{array}\right]$

In Problems 17–19, find the inverse, if there is one, of each matrix. If there is not an inverse, say that the matrix is singular.

$\left[\begin{array}{cc}4& -8\\ -1& 2\end{array}\right]$

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{r}3x-2y=1\\ 10x+10y=5\end{array}\right.$

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{l}5x-6y-3z=6\\ 4x-7y-2z=-3\\ 3x+y-7z=1\end{array}\right.$

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{r}2x+y+z=5\\ 4x-y-3z=1\\ 8x+y-z=5\end{array}\right.$

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{rl}x-2z& =1\\ 2x+3y& =-3\\ 4x-3y-4z& =3\end{array}\right.$

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{r}x-y+z=0\\ x-y-5z-6=0\\ 2x-2y+z-1=0\end{array}\right.$

In Problems 20–25, solve each system of equations using matrices. If the system has no solution, say that it is inconsistent.

$$\begin{gathered} \left\{x-y-z-t=1 \\ 2x+y-z+2t=3 \\ x-2y-2z-3t=0 \\ 3x-4y+z+5t=-3\right. \end{gathered}$$

find the value of each determinant

$\left|\begin{array}{cc}3& 4\\ 1& 3\end{array}\right|$

find the value of each determinant

$\left|\begin{array}{ccc}1& 4& 0\\ -1& 2& 6\\ 4& 1& 3\end{array}\right|\begin{array}{}\\ \\ \end{array}$

find the value of each determinant

$\left|\begin{array}{ccc}2& 1& -3\\ 5& 0& 1\\ 2& 6& 0\end{array}\right|$

Use Cramer’s Rule, if applicable, to solve each system.

$\left\{\begin{array}{l}x-2y=4\\ 3x+2y=4\end{array}\right.$

Use Cramer’s Rule, if applicable, to solve each system.

$\left\{\begin{array}{l}2x+3y-13=0\\ 3x-2y=0\end{array}\right.$

In Problems$29-31$, use Cramer’s Rule, if applicable, to solve each system.

$\left\{\begin{array}{l}x+2y-z=6\\ 2x-y+3z=-13\\ 3x-2y+3z=-16\end{array}\right.$

In Problems $32$ and$33$, use properties of determinants to find the value of each determinant if it is known that$\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|=8$

$\left|\begin{array}{cc}2x& y\\ 2a& b\end{array}\right|$

In Problems $32$and$33$, use properties of determinants to find the value of each determinant if it is known that $\left|\begin{array}{cc}x& y\\ a& b\end{array}\right|=8$

$\left|\begin{array}{cc}y& x\\ b& a\end{array}\right|$

In Problem, Write the partial fraction decomposition of each rational expression.

$\frac{6}{x(x-4)}$

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{ x-4}{x^2(x-1)}$

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x}{\left(x^2+9\right)(x+1)}$

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x^3}{{\left(x^2+4\right)}^2}$

In Problem, Write the partial fraction decomposition of each rational expression.
$\frac{x^2}{\left(x^2+1\right)(x^2-1)}$

In Problems 39–43, solve each system of equations.

$\left\{\begin{array}{l}2x+y+3=0\\ x^2+y^2=5\end{array}\right.$

In Problems 39–43, solve each system of equations.

$\left\{\begin{array}{l}2xy+y^2=10\\ 3y^2-xy=2\end{array}\right.$

In Problems 39–43, solve each system of equations.

$\left\{\begin{array}{l}{x}^2+y^2=6y\\ x^2=3y\end{array}\right.$

In Problems 39–43, solve each system of equations.

$\left\{\begin{array}{l}3x^2+4xy+5y^2=8\\ x^2+3xy+2y^2=0\end{array}\right.$

In Problems 39–43, solve each system of equations.

$\left\{\begin{array}{l}{x}^2-3x+y^2+y=-2\\ \frac{x^2-x}{y}+y+1=0\end{array}\right.$

In problems 39-43, Solve each system of equations 

$\left\{\begin{array}{l}{x}^2-3x+y^2+y=-2\\ \frac{x^2-x}{y}+y+1=0\end{array}\right.$

In Problems 44 and 45, graph each inequality by hand. Verify your results using a graphing utility.

$3x+4y\le 12$

In Problems 44 and 45, graph each inequality by hand. Verify your results using a graphing utility.

$y\le x^2$

In Problems 46–48, graph each system of inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points.

$\left\{\begin{array}{l}-2x+y\le 2\\ x+y\ge 2\end{array}\right.$

In Problems 46–48, graph each system of inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points.

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+y\le 4\\ 2x+3y\le 6\end{array}\right.$

In Problems 46–48, graph each system of inequalities by hand. Tell whether the graph is bounded or unbounded, and label the corner points.

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ 2x+y\le 8\\ x+2y\ge 2\end{array}\right.$

In Problems 49 and 50, graph each system of inequalities by hand.

$\left\{\begin{array}{l}{x}^2+y^2\le 16\\ x+y\ge 2\end{array}\right.$

In Problems 49 and 50, graph each system of inequalities by hand.

$\left\{\begin{array}{l}{x}^2+y^2\le 25\\ xy\le 4\end{array}\right.$

In Problems 51 and 52, solve each linear programming problem.

Maximize $z=3x+4y$ when $x\ge 0,y\ge 0,3x+2y\ge 6,x+y\le 8$

In Problems 51 and 52, solve each linear programming problem.

Minimize $z=3x+5y$ when $x\ge 0; y\ge 0; 3x+2y\le 12; x+3y\le 12$

Find A so that the system of equations has infinitely many
solutions.

$\left\{\begin{array}{l}2x+5y=5\\ 4x+10y=A\end{array}\right.$

Find A so that the system in Problem 53 is inconsistent.

Find the quadratic function $y=a{x}^2+bx+c$ that passes through the three points $(0,1),(1,0)$ and $(-2,1)$

A coffee distributor is blending a new coffee that will cost $\backslash (6.90$ per pound. It will consist of a blend of $\backslash )6.00$ per pound coffee and$\$9.00$  per pound coffee. What amounts of each type of coffee should be mixed to achieve the desired blend?

[Hint: Assume that the weight of the blended coffee is
$100$ pounds.]

A cookie company makes three kinds of cookies, oatmeal raisin, chocolate chip, and shortbread, packaged in small, medium, and large boxes. The small box contains $1$ dozen oatmeal raisin and $1$ dozen chocolate chip; the medium box has $2$ dozen oatmeal raisin, $1$ dozen chocolate chip, and $1$ dozen shortbread; the large box contains $2$ dozen oatmeal raisin, $2$ dozen chocolate chip, and $3$ dozen shortbread. If you require exactly $15$ dozen oatmeal raisin, $10$ dozen chocolate chip, and $11$ dozen shortbread, how many of each size box should you buy?

In Problems 1–4, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.

$\left\{\begin{array}{l}-2x+y=-7\\ 4x+3y=9\end{array}\right.$

In Problems 1–4, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l}\frac{1}{3}x-2y=1\\ 5x-30y=18\end{array}\right.$

In Problems 1–4, solve each system of equations using the method of substitution or the method of elimination. If the system has no solution, say that it is inconsistent.
$\left\{\begin{array}{l} x-y+2z=5\\ 3x+4y-z=-2\\ 5x+2y+3z=8\end{array}\right.$

solve each system of equations using the method

of substitution or the method of elimination. If the system has no

solution, say that it is inconsistent.

$\left\{\begin{array}{l}3x + 2y - 8z = -3\\ -x - \frac{2}{3}y + z = 1\\ 6x - 3y + 15z = 8\end{array}\right.$

Write the augmented matrix corresponding to the system of

equations:

$\left\{\begin{array}{l}4x - 5y + z = 0\\ -2x - y + 6 = -19\\ x + 5y - 5z = 10\end{array}\right.$

Write the system of equations corresponding to the

augmented matrix:

$\left[\begin{array}{cccc}3& 2& 4& -6\\ 1& 0& 8& 2\\ -2& 1& 3& -11\end{array}\right]$