Exam 8: Systems of Equations and Inequalities

Use a determinant and the given vertices of a triangle to find the area of the triangle. ​ ​

Use a graphing utility and Cramer's Rule to solve (if possible) the system of equations. ​ $\left\{ \begin{array} { r } x + 2 y - z = - 9 \\2 x - 2 y - 2 z = 0 \\- x + 3 y + 4 z = 0\end{array} \right.$ ​

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.) ​ $C = 3.7 \sqrt { x } + 5000 , R = 8.9 x$ ​

Solve the system of linear equations and check any solution algebraically. ​ $\left\{ \begin{aligned}5 x - 3 y + 2 z & = 9 \\2 x + 4 y - z & = 13 \\x - 11 y + 4 z & = 9\end{aligned} \right.$ ​

Write the matrix in reduced row-echelon form. ​ $\left[ \begin{array} { r r r } 4 & - 2 & 22 \\3 & 1 & - 6 \\3 & - 2 & 21\end{array} \right]$ ​

Find the maximum value of the objective function and where it occurs, subject to the constraints: ​ Objective function: ​ $z = x + 11 y$ ​ Constraints: ​ x \geq0 y \geq0 x+4y \leq20 x+y \leq18 2x+2y \leq21 ​

Solve the system graphically. ​ $\left\{ \begin{array} { r } x ^ { 2 } + y ^ { 2 } = 41 \\( x - 3 ) ^ { 2 } + y ^ { 2 } = 20\end{array} \right.$ ​

Find the consumer surplus and producer surplus. ​ Demand $p = 110 - 0.06 x$ ​ Supply $p = 45 + 0.2 x$ ​

A large region of forest has been infested with gypsy moths. The region is roughly triangular, as shown in the figure on the next page. From the northernmost vertex A of the region, the distances to the other vertices are x = 25 miles south and 10 miles east (for vertex B), and 20 miles south and 28 miles east (for vertex C). Use a graphing utility to approximate the number of square miles in this region. ​ ​

Use a graphing utility to solve the system of equations. Find the solution(s) accurate to two decimal places. ​ $\left\{ \begin{aligned}y & = e ^ { x + 2 } \\x - y + 3 & = 0\end{aligned} \right.$ ​

Find values of x, y, and λ that satisfy the system. These systems arise in certain optimization problems in calculus, and λ is called a Lagrange multiplier. ​ $\left\{ \begin{array} { r } 4 x - 4 x \lambda = 0 \\- 2 y + \lambda = 0 \\y - x ^ { 2 } = 0\end{array} \right.$ ​

According to automobile association of a country, on March 27, 2009, the national average price per gallon of regular unleaded (85-octane) gasoline was $2.09, and the price of premium unleaded (92-octane) gasoline was $2.24. Write an objective function that models the cost of the blend of mid-grade unleaded gasoline (90-octane). ​

Find x and y. ​ $\left[ \begin{array} { c c c c } 16 & 4 & 16 & 4 \\- 3 & 13 & 15 & 16 \\0 & 2 & 10 & 0\end{array} \right] = \left[ \begin{array} { c c c c } 16 & 4 & 7 x + 2 & 4 \\- 3 & 13 & 15 & 8 x \\0 & 2 & 3 y - 5 & 0\end{array} \right]$ ​

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

Find the sales necessary to break even (R = C) for the cost C of producing x units and the revenue R obtained by selling x units. (Round to the nearest whole unit.) ​ $C = 8610 x + 225,000$ , $R = 9860 x$ ​