Chapter 5

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Basedon the graph. (GRAPH CAN'T COPY) A hotel has 200 rooms. Those with kitchen facilities rent for \(\$ 100\) per night and those without kitchen facilities rent for \(\$ 80\) per night. On a night when the hotel was completely occupied, revenues were \(\$ 17,000\). How many of each type of room does the hotel have?

The points of intersection of the graphs of \(x y-20\) and \(x^{2}+y^{2}-41\) are joined to form a rectangle. Find the area of the rectangle.

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Based on the graph. (GRAPH CAN'T COPY) A new restaurant is to contain two-seat tables and four-seat tables. Fire codes limit the restaurant's maximum occupancy to 56 customers. If the owners have hired enough servers to handle 17 tables of customers, how many of each kind of table should they purchase?

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Based on the graph. (GRAPH CAN'T COPY) A rectangular lot whose perimeter is 360 feet is fenced along three sides. An expensive fencing along the lot's length costs \(\$ 20\) per foot and an inexpensive fencing along the two side widths costs only \(\$ 8\) per foot. The total cost of the fencing along the three sides comes to \(\$ 3280\). What are the lot's dimensions?

Solve the systems $$\left\\{\begin{array}{l} \log _{y} x-3 \\ \log _{y}(4 x)-5 \end{array}\right.$$

The figure shows the healthy weight region for various heights for people ages 35 and older. GRAPH CAN'T COPY If \(x\) represents height, in inches, and \(y\) represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$\left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right.$$ Use this information to solve Exercises 77-80. Is a person in this age group who is 6 feet tall weighing 205 pounds within the healthy weight region?

Use a system of linear equations to solve. Looking for Mr. Goodbar? It's probably not a good idea if you want to look like Mr. Universe or Julia Roberts. The graph shows the four candy bars with the highest fat content, representing grams of fat and calories in each bar. Based on the graph. (GRAPH CAN'T COPY) A rectangular lot whose perimeter is 320 feet is fenced along three sides. An expensive fencing along the lot's length costs 16 dollar per foot and an inexpensive fencing along the two side widths costs only 5 dollar per foot. The total cost of the fencing along the three sides comes to \(\$ 2140\). What are the lot's dimensions?

Solve the systems $$\left\\{\begin{array}{l} \log x^{2}-y+3 \\ \log x-y-1 \end{array}\right.$$

The figure shows the healthy weight region for various heights for people ages 35 and older. GRAPH CAN'T COPY If \(x\) represents height, in inches, and \(y\) represents weight, in pounds, the healthy weight region can be modeled by the following system of linear inequalities: $$\left\\{\begin{array}{l} 5.3 x-y \geq 180 \\ 4.1 x-y \leq 140 \end{array}\right.$$ Use this information to solve Exercises 77-80. Is a person in this age group who is 5 feet 8 inches tall weighing 135 pounds within the healthy weight region?

This will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$2 x-3 y-6$$

Many elevators have a capacity of 2000 pounds. a. If a child averages 50 pounds and an adult 150 pounds, write an inequality that describes when \(x\) children and \(y\) adults will cause the elevator to be overloaded. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

Use a system of linear equations to solve. When an airplane flies with the wind, it travels 800 miles in 4 hours. Against the wind, it takes 5 hours to cover the same distance. Find the plane's rate in still air and the rate of the wind.

This will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$f(x)=-\frac{2}{3} x$$

A patient is not allowed to have more than 330 milligrams of cholesterol per day from a diet of eggs and meat. Each egg provides 165 milligrams of cholesterol. Each ounce of meat provides 110 milligrams. a. Write an inequality that describes the patient's dietary restrictions for \(x\) eggs and \(y\) ounces of meat. b. Graph the inequality. Because \(x\) and \(y\) must be positive, limit the graph to quadrant I only. c. Select an ordered pair satisfying the inequality. What are its coordinates and what do they represent in this situation?

This will help you prepare for the material covered in the next section. In each exercise, graph the linear function. $$f(x)=-2$$

On your next vacation, you will divide lodging between large resorts and small inns. Let \(x\) represent the number of nights spent in large resorts. Let \(y\) represent the number of nights spent in small inns. a. Write a system of inequalities that models the following conditions: You want to stay at least 5 nights. At least one night should be spent at a large resort. Large resorts average \(\$ 200\) per night and small inns average \(\$ 100\) per night. Your budget permits no more than \(\$ 700\) for lodging. b. Graph the solution set of the system of inequalities in part (a). c. Based on your graph in part (b), what is the greatest number of nights you could spend at a large resort and still stay within your budget?

A person with no more than \(\$ 15,000\) to invest plans to place the money in two investments. One investment is high risk, high yield; the other is low risk, low yield. At least \(\$ 2000\) is to be placed in the high-risk investment. Furthermore, the amount invested at low risk should be at least three times the amount invested at high risk. Find and graph a system of inequalities that describes all possibilities for placing the money in the high-and low-risk investments.

What is a system of linear equations? Provide an example with your description.

Explain how to solve a system of equations using the substitution method. Use \(y-3=3 x\) and \(3 x+4 y=6\) to illustrate your explanation.

What is a linear inequality in two variables? Provide an example with your description.

Explain how to solve a system of equations using the addition method. Use \(3 x+5 y=-2\) and \(2 x+3 y=0\) to illustrate your explanation.

How do you determine if an ordered pair is a solution of an inequality in two variables, \(x\) and \(y?\)

When is it easier to use the addition method rather than the substitution method to solve a system of equations?

What is a half-plane?

When using the addition or substitution method, how can you tell if a system of linear equations has infinitely many solutions? What is the relationship between the graphs of the two equations?

What does a solid line mean in the graph of an inequality?

When using the addition or substitution method, how can you tell if a system of linear equations has no solution? What is the relationship between the graphs of the two equations?

What does a dashed line mean in the graph of an inequality?

Compare the graphs of \(3 x-2 y>6\) and \(3 x-2 y \leq 6\) Discuss similarities and differences between the graphs.

What is a system of linear inequalities?

Determine whether cach statement makes sense or does not make sense, and explain your reasoning. Even if a linear system has a solution set involving fractions, such as \(\left\\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\\}, 1\) can use graphs to determine if the solution set is reasonable.

What is a solution of a system of linear inequalities?

Explain how to graph the solution set of a system of inequalities.

What does it mean if a system of linear inequalities has no solution?

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$y \leq 4 x+4$$

Write a system of equations having \([(-2,7)]\) as a solution set. (More than one system is possible.)

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$y \geq \frac{2}{3} x-2$$

Solve the system for \(x\) and \(y\) in terms of \(a_{1}, b_{1}, c_{1}, a_{2}, b_{2}\) and \(c_{2}\) : $$\left\\{\begin{array}{l}a_{1} x+b_{1} y=c_{1} \\ a_{2} x+b_{2} y=c_{2}\end{array}\right.$$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$y \geq \frac{1}{2} x^{2}-2$$

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$2 x+y \leq 6$$

The group should write four different word problems that can be solved using a system of linear equations in two variables. All of the problems should be on different topics. The group should turn in the four problems and their algebraic solutions.

Graphing utilities can be used to shade regions in the rectangular coordinate system, thereby graphing an inequality in two variables Read the section of the user's manual for your graphing utility that describes how to shade a region. Then use your graphing utility to graph the inequalities in Exercises 97-102. $$3 x-2 y \geq 6$$

If \(x=3, y=2,\) and \(z=-3,\) does the ordered triple \((x, y, z)\) satisfy the equation \(2 x-y+4 z=-8 ?\)

Write an equation involving \(a, b,\) and \(c\) based on the following description: When the value of \(x\) in \(y=a x^{2}+b x+c\) is \(4,\) the value of \(y\) is \(1682 .\)

In Exercises 106-109, determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing a linear inequality, I should always use \((0,0)\) as a test point because it's easy to perform the calculations when 0 is substituted for each variable.

In Exercises 106-109, determine whether each statement makes sense or does not make sense, and explain your reasoning. When graphing \(3 x-4 y<12,\) it's not necessary for me to graph the linear equation \(3 x-4 y-12\) because the inequality contains a \(<\) symbol, in which equality is not included.

Write a system of inequalities whose solution set includes every point in the rectangular coordinate system.

Sketch the graph of the solution set for the following system of inequalities: $$\left\\{\begin{array}{l} y \geq n x+b(n<0, b>0) \\ y \leq m x+b(m>0, b>0). \end{array}\right.$$

Exercises 116-118 will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{aligned} x+y & \geq 6 \\ x & \leq 8 \\ y & \geq 5 \end{aligned}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(3 x+2 y\) at each of the points obtained in part (b).

Exercises 116-118 will help you prepare for the material covered in the next section. a. Graph the solution set of the system: $$\left\\{\begin{aligned} x & \geq 0 \\ y & \geq 0 \\ 3 x-2 x & \leq 6 \\ y & \leq-x+7 .\end{aligned}\right.$$ b. List the points that form the corners of the graphed region in part (a). c. Evaluate \(2 x+5 y\) at each of the points obtained in part (b).