Systems of Equations and Inequalities

use the given matrices to compute each

expression.

$$\begin{gathered} A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& -2\end{array}\right] \\ B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right] \\ C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right] \end{gathered}$$

$2A+C$

use the given matrices to compute each

expression.

$$\begin{gathered} A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& -2\end{array}\right] \\ B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right] \\ C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right] \end{gathered}$$

$A-3C$

use the given matrices to compute each

expression.

$$\begin{gathered} A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& -2\end{array}\right] \\ B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right] \\ C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right] \end{gathered}$$

$CB$

use the given matrices to compute each

expression.

$$\begin{gathered} A=\left[\begin{array}{cc}1& -1\\ 0& -4\\ 3& -2\end{array}\right] \\ B=\left[\begin{array}{ccc}1& -2& 5\\ 0& 3& 1\end{array}\right] \\ C=\left[\begin{array}{cc}4& 6\\ 1& -3\\ -1& 8\end{array}\right] \end{gathered}$$

$BA$

find the inverse of each nonsingular matrix.

$\left[\begin{array}{cc}3& 2\\ 5& 4\end{array}\right]$

find the inverse of each nonsingular matrix.

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

solve each system of equations using matrices.

If the system has no solution, say that it is inconsistent.

$$\begin{gathered} 6x+3y=12 \\ 2x-y=-2 \end{gathered}$$

solve each system of equations using matrices.

If the system has no solution, say that it is inconsistent.

$$\begin{gathered} x+\frac{1}{4}y=7 \\ 8x+2y=56 \end{gathered}$$

solve each system of equations using matrices.

If the system has no solution, say that it is inconsistent.

$$\begin{gathered} x+2y+4z=-3 \\ 2x+7y+15z=-12 \\ 4x+7y+13z=-10 \end{gathered}$$

solve each system of equations using matrices.

If the system has no solution, say that it is inconsistent.

$$\begin{gathered} 2x+2y-3z=5 \\ x-y+2z=8 \\ 3x+5y-8z=-2 \end{gathered}$$

find the value of each determinant.

$\left|\begin{array}{cc}-2& 5\\ 3& 7\end{array}\right|$

find the value of each determinant.

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

A store that specializes in selling nuts has available $72$ pounds (lb) of cashews and $120$ lb of peanuts. These are to be mixed in $12$ ounce (oz) packages as follows: a lower-priced package containing $8$ oz of peanuts and $4$ oz of cashews and a quality package containing $6$ oz of peanuts and $6$ oz of cashews.

(a) Use x to denote the number of lower-priced packages, and use y  to denote the number of quality packages. Write a system of linear inequalities that describes the possible number of each kind of package.
(b) Graph the system and label the corner points.

On a recent trip to the Cuyabeno Wildlife Reserve in the Amazon region of Ecuador, Mike took a $100$-kilometer trip by speedboat down the Aguarico River from Chiritza to the Flotel Orellana. As Mike watched the Amazon unfold, he wondered how fast the speedboat was going and how fast the current of the white water Aguarico River was. Mike timed the trip downstream at $2.5$ hours and the return trip at $3$ hours. What were the two speeds?

If Bruce and Bryce work together for $1$ hour and $20$ minutes, they will finish a certain job. If Bryce and Marty work together for $1$ hour and $36$ minutes, the same job can be finished. If Marty and Bruce work together, they can complete this job in $2$ hours and $40$ minutes. How long will it take each of them working alone to finish the job?

A factory produces gasoline engines and diesel engines. Each week the factory is obligated to deliver at least $20$ gasoline engines and at least $15$ diesel engines. Due to physical limitations, however, the factory cannot make more than $60$ gasoline engines nor more than $40$  diesel engines in any given week. Finally, to prevent layoffs, a total of at least $50$ engines must be produced. If gasoline engines cost $\backslash (450$ each to produce and diesel engines cost $\backslash )550$ each to produce, how many of each should be produced per week to minimize the cost? What is the excess capacity of the factory; that is, how many of each kind of engine is being produced in excess of the number that the factory is obligated to deliver?

Describe four ways of solving a system of three linear equations containing three variables. Which method do you prefer? Why?

In Problems $1-6$, solve each equation.

$2x^2-x=0$.

In Problems $1-6$, solve each equation.

$\sqrt{3x+1}=4$.

In Problems $1-6$, solve each equation.

$2x^3-3x^2-8x-3=0$.

In Problems $1-6$, solve each equation.

$3^x=9^{x+1}$.

In Problems $1-6$, solve each equation.

${\log }_3\left(x-1\right)+{\log }_3\left(2x+1\right)=2$.

In Problems $1-6$, solve each equation.

$3^x=e$.

Determine whether the function $g\left(x\right)=\frac{2x^3}{x^4+1}$ is even, odd, or neither. Is the graph of $g$ symmetric with respect to the $x$-axis, $y$-axis, or origin?

Graph $f\left(x\right)=3^{x-2}+1$ using transformations. What is the domain, range, and horizontal asymptote of $f$?

The function $f\left(x\right)=\frac{5}{x+2}$ is one to one. Find $f^{-1}$. Find the domain and the range of $f$ and the domain and the range of $f^{-1}$.

Graph each equation.

$\begin{array}{l}\left(a\right)y=3x+6 \left(b\right)x^2+y^2=4 \\ \left(c\right)y=x^3 \left(d\right)y=\frac{1}{x}\\ \left(e\right)y=\sqrt{x} \left(f\right)y=e^x\\ \left(g\right)y=\ln x \left(h\right)2x^2+5y^2=1\\ \left(i\right)x^2-3y^2=1 \left(j\right)x^2-2x-4y+1=0\end{array}$

$f\left(x\right)=x^3-3x+5$.

(a) Using a graphing utility, graph $f$ and approximate the zero(s) of $f$.

(b) Using a graphing utility, approximate the local maxima and local minima.

(c) Determine the intervals on which $f$ is increasing.

In Problems $19$ and $20$, use Cramer’s Rule, if possible, to solve each system

$\left\{\begin{array}{l}4x+3y=-23\\ 3x-5y=19\end{array}\right.$

In Problems $19$ and $20$, use Cramer’s Rule, if possible, to solve each system.

$\left\{\begin{array}{rr}4x-3y+2z=& 15\\ -2x+y-3z=& -15\\ 5x-5y+2z=& 18\end{array}\right.$

In Problems $21$ and $22$, solve each system of equations.

$\left\{\begin{array}{r}3x^2+y^2=12\\ y^2=9x\end{array}\right.$

In Problems $21$ and $22$, solve each system of equations.

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

Graph the system of inequalities: 

$\left\{\begin{array}{l}{x}^2+y^2\le 100\\ 4x-3y\ge 0\end{array}\right.$

In Problems $24$ and $25$, write the partial fraction decomposition of each rational expression.

$\frac{3x+7}{(x+3{)}^2}$

In Problems $24$ and $25$, write the partial fraction decomposition of each rational expression.

$\frac{4x^2-3}{x{\left(x^2+3\right)}^2}$

Graph the system of inequalities. Tell whether the graph is bounded or unbounded, and label all corner points. 

$\left\{\begin{array}{l}x\ge 0\\ y\ge 0\\ x+2y\ge 8\\ 2x-3y\ge 2\end{array}\right.$

Maximize $z=5 x+8 y$ subject to $x\ge 0,2x+y\le 8$ and $x-3y\le -3$

Megan went clothes shopping and bought $2$ pairs of flare jeans, $2$ camisoles, and $4$ T-shirts for $\backslash (90.00$. At the same store, Paige bought one pair of flare jeans and $3$ T-shirts for $\backslash )42.50$, while Kara bought $1$ pair of flare jeans, $3$ camisoles, and $2$ T-shirts for $\$62.00$. Determine the price of each clothing item.