Exam 6: Systems of Equations and Inequalities

The federal debt of the United States as a percentage of the Gross Domestic Product (GDP) from 2001 to 2005 is shown in the table. In the table, $x$ represents the year, with $x = 0$ corresponding to 2002. Year, x \% of GDP -1 57.4 0 59.7 1 62.6 2 63.7 3 64.3 Find the least squares regression parabola $y = a x ^ { 2 } + b x + c$ for the data by solving the following system. $\left\{ \begin{aligned}5 c + 5 b + 15 a & = 307.7 \\5 c + 15 b + 35 a & = 325.5 \\15 c + 35 b + 99 a & = 953.5\end{aligned} \right.$ Use the model to predict the federal debt as percents of GDP in 2006. Round to the nearest tenth percents.

Graph the solution set of the system of inequalities below. $\left\{ \begin{array} { l } y \leq - x + 2 \\y \geq - x - 2 \\y \geq 0 \\y \leq 2\end{array} \right.$

Solve the system by the method of substitution. $\left\{ \begin{array} { l } y = x ^ { 3 } + 2 x ^ { 2 } - 1 \\y = x ^ { 2 } + 12 x - 1\end{array} \right.$

Find the maximum value of the objective function and where it occurs, if one exists. Objective function: $z = 2 x - 4 y$ Constraints: x \geq0 y \geq0 -x+y \leq1 -5x+y \geq5

Determine which ordered pair is a solution of the system. $\left\{ \begin{array} { c } x + 4 y ^ { 2 } = 28 \\- x + y = 5\end{array} \right.$

Solve the system of equations below: $- 6 x + y + 7 z = 20$ $7 x - 2 y - 9 z = - 25$

Solve the system of equations below by the substitution method. $5 x + 8 y = 13$ $- 5 x - 8 y = - 11$

Determine which one of the ordered triples below is a solution of the given system of equations. $\left\{ \begin{array} { l } 6 x + 3 y + 3 z = - 81 \\6 x - y - z = - 21 \\5 x - 5 y + 5 z = - 25\end{array} \right.$

Determine whether the system of equations below has one solution, two solutions, or no solution. $\left\{ \begin{array} { l } y = x ^ { 2 } - 16 x + 11 \\y = - x ^ { 2 } - 8 x - 11\end{array} \right.$

Solve the system graphically. $\left\{ \begin{array} { l } 3 x + 4 y = 4 \\y ^ { 2 } - x ^ { 2 } = 9\end{array} \right.$

Use a graphing utility to graph the inequality. Shade the region representing the solution. $y > 3 - \frac { 2 } { 5 } x$

Solve the system of linear equations. $\left\{ \begin{aligned}4 x + 3 y - z & = - 20 \\x + 3 z & = - 2 \\2 x + 4 y + 3 z & = - 20\end{aligned} \right.$

Solve the system by the method of elimination. $\left\{ \begin{array} { c } \frac { 1 } { 4 } x + \frac { 1 } { 8 } y = - \frac { 5 } { 8 } \\2 x + y = - 5\end{array} \right.$

Maximize the object function $z = 15 x + y$ subject to the constraints $x + 5 y \leq 25 , x + y \leq 9,4 x + 2 y \leq 32 , x \geq 0 \text {, and } y \geq 0$ .

Maximize the object function $z = 10 x + 8 y$ subject to the constraints $x + 5 y \leq 25 , x + y \leq 9,4 x + 2 y \leq 32 , x \geq 0 \text {, and } y \geq 0$ .

A company has budgeted a maximum of $1,200,000 for national advertising an allergy medication. Each minute of television time costs $100,000 and each one-page newspaper ad costs $40,000. Each television ad is expected to be viewed by 43 million viewers, and each newspaper ad is expected to be seen by 18 million readers. What is the optimal amount that should be spent on advertising for each type ad?

Sketch the graph of the inequality below. $x + y > 3$

You plan to invest up to $60,000 in two different interest-bearing accounts. Each account is to contain at least $8000. Moreover, one account should have at least three times the amount that is in the other account. Find a system of inequalities that describes the amount that you can invest in each account.

Solve the system of equations below, if possible. $\left\{ \begin{array} { c } 4 x + 2 y + 5 z = - 3 \\x + y + z = 0 \\3 x + y + 4 z = 1\end{array} \right.$

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.) $C = 8.8 \sqrt { x } + 2000 \quad R = 7.9 x$