Exam 7: Systems Of Equations and Inequalities

Find values of x, y, and $\lambda$ that satisfy the system.These systems arise in certain optimization problems in calculus, and $\lambda$ is called a Lagrange multiplier. $\left\{\begin{array}{ll}y+\lambda & =0 \\x+\lambda & =0 \\x+y-18 & =0\end{array}\right.$

Find the sales necessary to break even (R - C = 0) for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit.) $C = 6.7 \sqrt { x } + 3000 , R = 8.1 x$

Find the equation of the circle $x ^ { 2 } + y ^ { 2 } + D x + E y + F = 0$ that passes through the points. $( 0,0 ) , ( 6,6 ) , ( 12,0 )$

What are the dimensions of a rectangular tract of land if its perimeter is 50 kilometers and its area is 150 square kilometers? ​

Solve the system of linear equations by the method of elimination.Find (r,s) and check your solution algebraically. $\left\{ \begin{array} { c } 2 r + 4 s = 7 \\16 r + 50 s = 77\end{array} \right.$

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

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. $( - 1,4 ) , ( 0,4 ) , ( 1,0 )$

Solve the system graphically. $\left\{ \begin{array} { r } - x + 2 y = 3 \\x - y = 2\end{array} \right.$

The sales of various types of lawn and garden tools vary according to the season.At a certain home improvement store, the monthly sales H of garden hoes (hoes sold per month) decline from July to October whereas the monthly sales of lawn rakes R (rakes sold per month) increase during this same interval.The sales of these two items during the calendar months July-October are modeled by the equations: H(t)=64-6t R(t)=17t-143 where t is the month (t = 7 corresponds to July).In which month does the number of rakes sold equal the number of hoes sold

Solve the system of linear equations by the method of elimination, find (x,y). $\left\{ \begin{array} { c } \frac { 9 } { 4 } x + \frac { 1 } { 4 } y = \frac { - 7 } { 4 } \\9 x + y = - 7\end{array} \right.$

A total of $23,000 is invested in two corporate bonds that pay 3.5% and 5% simple interest.The investor wants an annual interest income of $880 from the investments.What amount should be invested in the 3.5% bond? ​

Solve the system of linear equations and check any solution algebraically. $\left\{ \begin{array} { l l } 2 x + y - z & = 5 \\x - 2 y + 2 z & = - 7 \\3 x - y + z & = 3\end{array} \right.$

Find the minimum and maximum values of the objective function and where it occurs, subject to the indicated constraints. Objective function: Z = 3x + 4y Constraints: $\left\{ \begin{array} { l } y \geq 0 \\x - y \geq - 2 \\2 x + 3 y \geq 6 \\5 x + 2 y \leq 25\end{array} \right.$

Solve the system graphically. $\left\{ \begin{aligned}x + y & = 0 \\3 x - 2 y & = 10\end{aligned} \right.$

Write the form of the partial fraction decomposition of the rational expression. $\frac { 6 x - 1 } { x ( x - 4 ) }$

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

Find the minimum value of the objective function and where it occurs, subject to the constraints: Objective function: Z = 8x + y Constraints: X $\ge$ 0 Y $\ge$ 0 X + 4y $\le$ 20 X + y $\le$ 18 2x + 2y $\le$ 21

Solve the system graphically. $\left\{ \begin{aligned}x - 3 y & = - 7 \\5 x + 3 y & = - 8\end{aligned} \right.$

Use back-substitution to solve the system of linear equations. $\left\{ \begin{array} { c c c } - 3 x - 9 y - 7 z & = 0 \\4 y + 2 z & = - 8 \\z & = & 6\end{array} \right.$

A store sells two models of laptop computers.Because of the demand, the store stocks at least twice as many units of model A as of model B.The costs to the store for the two models are $800 and $1200, respectively.The management does not want more than $29,000 in computer inventory at any one time, and it wants at least four model A laptop computers and two model B laptop computers in inventory at all times.Find and graph a system of inequalities describing all possible inventory levels.