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. $\left\{ \begin{array} { c } 5 x - 5 x \lambda = 0 \\- 2 y + \lambda = 0 \\y - x ^ { 2 } = 0\end{array} \right.$

Find the maximum value of the objective function and where it occurs, subject to the constraints: Objective function: Z = 12x + 24y Constraints: X $\ge$ 0 Y $\ge$ 0 X + 4y $\le$ 20 X + y $\le$ 18 2x + 2y $\le$ 21

Write the partial fraction decomposition of the rational expression.Check your result algebraically. $\frac { 4 x ^ { 2 } + 3 x + 20 } { \left( x ^ { 2 } + 5 \right) ^ { 2 } }$

The linear programming problem has an unusual characteristic.Select a graph of the solution region for the problem and describe the unusual characteristic.Find the maximum value of the objective function (if possible) and where it occurs. ​ Objective function: ​ Z = x + y ​ Constraints: ​ X ≥ 0 Y ≥ 0 -x + y ≤ 1 -x + 5y ≤ 7 ​

Select the correct graph of the inequality. $x \geq 3$

Write the partial fraction decomposition of the rational expression. $\frac { 1 } { a ^ { 2 } - x ^ { 2 } }$

Select an inequality for the shaded region shown in the figure.

Write the partial fraction decomposition of the rational function. $\frac { x - 16 } { x ( x - 4 ) }$

Solve the system of linear equations and check any solution algebraically. $\left\{ \begin{array} { c } x + 4 z = 1 \\x + y + 10 z = 10 \\2 x - y + 2 z = - 5\end{array} \right.$

Determine which one of the ordered triples below is a solution of the given system of equations. $\left\{ \begin{array} { l l } 5 x + 7 y - 7 z & = 11 \\2 x - 3 y - z & = - 18 \\3 x - 4 y + z & = - 41\end{array} \right.$

Select the correct graph of the inequality. $4 y ^ { 2 } - x < 0$

Find the consumer surplus and producer surplus. Demand $p = 100 - 0.03 x$ Supply $p = 35 + 0.1 x$

Select the correct graph of the inequality. $6 \geq y$

Find the equation of the parabola $y = a x ^ { 2 } + b x + c$ that passes through the points. $( 0,5 ) , ( 1,6 ) , ( 2,5 )$

Select the ordered pair that is a solution of the system of equations. $\left\{ \begin{aligned}- \log x + 3 & = 2 \\\frac { 1 } { 9 } x + y & = \frac { 28 } { 9 }\end{aligned} \right.$

Find the equilibrium point (x,p) of the demand and supply equations.The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations. Demand Supply p=410-0.0004x p=230+0.0002x

The perimeter of a triangle is 110 feet.The longest side of the triangle is 21 feet longer than the shortest side.The sum of the lengths of the two shorter sides is 12 feet more than the length of the longest side.Find the lengths of the sides of the triangle. ​

Find the maximum value of the objective function and where it occurs, subject to the indicated constraints. ​ Objective function: ​ Z = 4x + 3y ​ Constraints: ​ X ≥ 0 Y ≥ 0 X + y ≤ 4​ ​​ ​

Solve the system of linear equations by the method of elimination.Find (x,y) and check your solution algebraically. $\left\{ \begin{array} { c } 5 x + 3 y = 6 \\3 x - y = 5\end{array} \right.$

Write the partial fraction decomposition of the rational expression.Check your result algebraically. $\frac { 1 } { 25 x ^ { 2 } - 36 }$