Exam 5: Systems of Equations and Inequalities

Solve the problem. -A large department store needs at least 3,200 labor hours covered per week. It employs full-time staff 40 hr/wk and part-time staff 30 hr/wk. The cost to employ a full-time staff member is more Because the company pays benefits such as health care and life insurance. Hours per Week Cost per Hour Full-time 40 20 Part-time 30 13 The store manager also knows that to make the store run efficiently, the number of full-time employees must Be at least 1.25 times the number of part-time employees. Determine the number of full-time Employees and the number of part-time employees that should be used to minimize the weekly labor Cost. What is the minimum weekly cost to staff the store under these conditions.

Solve the system of equations by using the addition method. -Solve the system by the addition method. $7.7 x + 7.1 y = - 3.5$ $- 7.1 x - 7.3 y = 8.5$

Solve the problem. -A furniture manufacturer builds tables. The cost for materials and labor to build a kitchen table is $270 and the profit is $180. The cost to build a dining room table is $300 and the profit is $220. Let X represent the number of kitchen tables produced per month. Let y represent the number of dining room Tables produced per month. The manufacturing process is subject to the following constraints. 1) The number of each type of table cannot be negative. 2) Due to labor and equipment restrictions, the company can build at most 110 kitchen tables. 3) The company can build at most 70 dining room tables. 4) The company does not want to exceed a monthly cost of $48,000. a. Write an objective function representing the monthly profit for producing and selling $x$ kitcher tables and y dining room tables. b. How many kitchen tables and how many dining room tables should be produced to maximize profit? (Assume that all tables produced will be sold.)

Solve the problem. -In an open market, the price of an item is dependent on the demand by consumers and the supply offered by producers. Competition between buyers and sellers steers the price toward an equilibrium price--that is, The price where supply equals demand. The number of items offered and sold at the equilibrium price is the Equilibrium quantity. The price $p$ (in $\$$ ) of a cookbook is determined by the number of cookbooks $x$ demanded by consume and supplied by the publisher. Supply: $p = 0.003 x$ Demand: $p = - 0.005 x + 80$ a. Solve the system of equations defined by the supply and demand models. b. What is the equilibrium price? c. What is the equilibrium quantity?

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -For a constant real number k, the inequality y > k represents the half-plane (above/below) the (horizontal/vertical) line y = k.

Solve the problem. -Recall that an equation of a circle can be written in the form $( x - h ) ^ { 2 } + ( y - k ) ^ { 2 } = r ^ { 2 }$ , where $( h , k )$ is the center and $r$ is the radius. Expanding terms, the equation can also be written in the form $x ^ { 2 } + y ^ { 2 }$ $+ A x + B y + C = 0$ . a . Find an equation of the form $x ^ { 2 } + y ^ { 2 } + A x + B y + C = 0$ that represents the circle that passes through the three points $( 4 , - 4 ) , ( 2 , - 8 )$ , and $( - 5 , - 1 )$ . b. Find the center and radius of the circle.

Find the partial fraction decomposition. (Hint: Use the rational zero theorem to factor the denominator.) - $\frac { 5 x ^ { 2 } + 11 x + 8 } { x ^ { 3 } + 3 x ^ { 2 } + 3 x + 1 }$

A________ to a system of equations in two variables is an ordered pair that is a solution to each individual equation in the system.

Find the constants A and B so that the two polynomials are equal. (Hint: Create a system of linear equations by equating the constant terms and by equating the coefficients on the x terms and $x ^ { 2 }$ terms.) - $- 16 x ^ { 2 } + 10 x - 6 = - 2 A x ^ { 2 } - A x + 5 A - 6 B x ^ { 2 } + 6 B x - 3 B + 6 C x ^ { 2 } + 3 C x - 5 C$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -A solution to a linear equation in three variables is an ordered________ (x, y, z) that satisfiesb the equation.

Graph the solution set. - x+y <4 0\leqy \leq5 x \geq-2

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -An equation of the form $A x + B y + C z = D$ (where A, B, and C are not all zero) is called a________equation in three variables.

Solve the system by using any method. - +(y-4=16 y=-+8

Find the partial fraction decomposition. [Hint: Use the substitution $u = e ^ { x }$ and recall that $\left. e ^ { 2 x } = \left( e ^ { x } \right) ^ { 2 } \cdot \right]$ - $\frac { - 8 e ^ { x } + 6 } { e ^ { 2 x } - 5 e ^ { x } - 6 }$

Solve the system. If a system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent. - 3x+2y-6z =-15 -3y+6z =21 4z =32

Choose the one alternative that best completes the statement or answers the question. Set up the form for the partial fraction decomposition. Do not solve for A, B, C, and so on. - $\frac { 10 } { x ^ { 2 } - 6 x }$

A system of linear equations in two variables may have no solution. In such a case, the equations represent________ lines.

Solve the system by substitution. - 16x-y=7 8x+5y=31

Solve the problem. -To reach a sales meeting, Carrie traveled by plane for 2 hours and by car for an additional 2 hours. The plane travels 54 mph faster than the car. If the total distance traveled is 308 miles, what are the Speeds of the plane and the car? (Assume that the plane and the car travel at constant rates.)

Choose the one alternative that best completes the statement or answers the question. Determine whether the ordered pair is a solution to the inequality. - $| x | - 1 > 1$