Deck 8: Systems of Equations and Inequalities

Question 1

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.)
​
$C = 8610 x + 225,000$ ,
$R = 9860 x$
​

Answer 1

180 units

Question 2

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

Answer 2

x = 2, y = 5

Question 3

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

Answer 3

$$\frac { 1 } { 2 a } \left( \frac { 1 } { a + x } + \frac { 1 } { a - x } \right)$$

Question 4

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

Question 5

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

Question 6

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.
​

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Question 7

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

Question 8

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

Question 9

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

Question 10

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.
​ $$\left\{ \begin{array} { r } 4 x - 4 x \lambda = 0 \\- 2 y + \lambda = 0 \\y - x ^ { 2 } = 0\end{array} \right.$$ ​

Question 11

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

Question 12

Find the maximum value of the objective function and where it occurs, subject to the constraints:
​
Objective function:
​ $$z = x + 11 y$$ ​
Constraints:
​ $\begin{aligned}x & \geq 0 \\y & \geq 0 \\x + 4 y & \leq 20 \\x + y & \leq 18 \\2 x + 2 y & \leq 21\end{aligned}$ ​

Question 13

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

Question 14

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).
​

Question 15

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