Chapter 7

In Exercises 55-56, find the value of \(y\) if the line through the two given points is to have the indicated slope. \((3, y)\) and \((1,4), m=-3\)

Each group member should consult an almanac, newspaper, magazine, or the Internet to find data that can be modeled by linear, exponential, logarithmic, or quadratic functions. Group members should select the two sets of data that are most interesting and relevant. Then consult a person who is familiar with graphing calculators to show you how to obtain a function that best fits each set of data. Once you have these functions, each group member should make one prediction based on one of the models, and then discuss a consequence of this prediction. What factors might change the accuracy of the prediction?

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

Find the value of \(y\) if the line through the two given points is to have the indicated slope. \((-2, y)\) and \((4,-4), m=\frac{1}{3}\)

Describe how to solve a system of linear inequalities.

Exercises 57-60 describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. A company that manufactures small canoes has a fixed cost of \(\$ 18,000\). It costs \(\$ 20\) to produce each canoe. The selling price is \(\$ 80\) per canoe. (In solving this exercise, let \(x\) represent the number of canoes produced and sold.)

Make Sense? In Exercises 58-61, 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.

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. A company that manufactures bicycles has a fixed cost of \(\$ 100,000\). It costs \(\$ 100\) to produce each bicycle. The selling price is \(\$ 300\) per bike. (In solving this exercise, let \(x\) represent the number of bicycles produced and sold.)

Make Sense? 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.

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invest in a new play. The cost includes an overhead of \(\$ 30,000\), plus production costs of \(\$ 2500\) per performance. A sold-out performance brings in \(\$ 3125\). (In solving this exercise, let \(x\) represent the number of sold- out performances.)

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. Systems of linear inequalities are appropriate for modeling healthy weight because guidelines give healthy weight ranges, rather than specific weights, for various heights.

Describe a number of business ventures. For each exercise, a. Write the cost function, \(C\). b. Write the revenue function, \(R\). c. Determine the break-even point. Describe what this means. You invested \(\$ 30,000\) and started a business writing greeting cards. Supplies cost 2 cents per card and you are selling each card for 50 cents. (In solving this exercise, let \(x\) represent the number of cards produced and sold.)

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. I graphed the solution set of \(y \geq x+2\) and \(x \geq 1\) without using test points.

An important application of systems of equations arises in connection with supply and demand. As the price of a product increases, the demand for that product decreases. However, at higher prices, suppliers are willing to produce greater quantities of the product. The price at which supply and demand are equal is called the equilibrium price. The quantity supplied and demanded at that price is called the equilibrium quantity. Exercises 61-62 involve supply and demand. The table shows the price of a gallon of unleaded premium gasoline. For each price, the table lists the number of gallons per day that a gas station sells and the number of gallons per day that can be supplied. $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Price per } \\ \text { Gallon } \end{array} & \begin{array}{c} \text { Gallons Demanded } \\ \text { per Day } \end{array} & \begin{array}{c} \text { Gallons Supplied } \\ \text { per Day } \end{array} \\ \hline \$ 3.20 & 1400 & 200 \\ \hline \$ 3.60 & 1200 & 600 \\ \hline \$ 4.40 & 800 & 1400 \\ \hline \$ 4.80 & 600 & 1800 \\ \hline \end{array} $$ The data in the table are described by the following demand and supply models: Demand Model Supply Model \(p=-0.002 x+6\) $$ p=0.001 x+3 . $$ a. Solve the system and find the equilibrium quantity and the equilibrium price for a gallon of unleaded premium gasoline. b. Use your answer from part (a) to complete this statement: If unleaded premium gasoline is sold for per gallon, there will be a demand for gallons per day and gallons will be supplied per day.

In Exercises 61-62, find a linear function in slope-intercept form that models the given description. Each function should model the percentage of total spending, \(p(x)\), by Americans \(x\) years after 1950 . In 1950 , Americans spent \(22 \%\) of their budget on food. This has decreased at an average rate of approximately \(0.25 \%\) per year since then.

The table shows the price of a package of cookies. For each price, the table lists the number of packages that consumers are willing to buy and the number of packages that bakers are willing to supply. $$ \begin{array}{|c|c|c|} \hline \begin{array}{c} \text { Price of a } \\ \text { Package } \\ \text { of Cookies } \end{array} & \begin{array}{c} \text { Quantity Demanded } \\ \text { (millions of packages) } \\ \text { per Week } \end{array} & \begin{array}{c} \text { Quantity Supplied } \\ \text { (millions of packages) } \\ \text { per Week } \end{array} \\ \hline 30 \varsigma & 150 & 70 \\ \hline 40 \% & 130 & 90 \\ \hline 60 c & 90 & 130 \\ \hline 6 \text { e } 70 \% \text { c } & 70 & 150 \\ \hline \end{array} $$ The data in the table can be described by the following demand and supply models: $$ \begin{array}{cc} \text { Demand Model } & \text { Supply Model } \\ p=-0.5 x+105 & p=0.5 x-5 . \end{array} $$ a. Solve the system and find the equilibrium quantity and the equilibrium price for a package of cookies. b. Use your answer from part (a) to complete this statement: If cookies are sold for per package, there will be a demand for million packages per week and bakers will supply million packages per week.

Find a linear function in slope-intercept form that models the given description. Each function should model the percentage of total spending, \(p(x)\), by Americans \(x\) years after 1950 . In 1950 , Americans spent \(3 \%\) of their budget on health care. This has increased at an average rate of approximately \(0.22 \%\) per year since then.

Without graphing, in Exercises 64-67, determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}3 x+y<9 \\ 3 x+y>9\end{array}\right.\)

Harsh, mandatory minimum sentences for drug offenses account for more than half the population in U.S. federal prisons. The bar graph shows the number of inmates in federal prisons, in thousands, for drug offenses and all other crimes in 1998 and 2010. (Other crimes include murder, robbery, fraud, burglary, weapons offenses, immigration offenses, racketeering, and perjury.) a. In 1998 , there were 60 thousand inmates in federal prisons for drug offenses. For the period shown by the graph, this number increased by approximately \(2.8\) thousand inmates per year. Write a function that models the number of inmates, \(y\), in thousands, for drug offenses \(x\) years after \(1998 .\) b. In 1998 , there were 44 thousand inmates in federal prisons for all crimes other than drug offenses. For the period shown by the graph, this number increased by approximately \(3.8\) thousand inmates per year. Write a function that models the number of inmates, \(y\), in thousands, for all crimes other than drug offenses \(x\) years after \(1998 .\) c. Use the models from parts (a) and (b) to determine in which year the number of federal inmates for drug offenses was the same as the number of federal inmates for all other crimes. How many inmates were there for drug offenses and for all other crimes in that year?

Without graphing, Determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}6 x-y \leq 24 \\ 6 x-y>24\end{array}\right.\)

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

Describe how to find the \(x\)-intercept of a linear equation.

Without graphing, Determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}3 x+y \leq 9 \\ 3 x+y \geq 9\end{array}\right.\)

Describe how to find the \(y\)-intercept of a linear equation.

Without graphing, Determine if each system has no solution or infinitely many solutions. \(\left\\{\begin{array}{l}6 x-y \leq 24 \\ 6 x-y \geq 24\end{array}\right.\)

Explain how to solve a system of equations using graphing.

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.

Describe how to calculate the slope of a line passing through two points.

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.

Describe how to graph a line using the slope and \(y\)-intercept. Provide an original example with your description.

What is the disadvantage to solving a system of equations using the graphing method?

What does it mean if the slope of a line is 0 ?

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

What does it mean if the slope of a line is undefined?

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

What is the least number of points needed to graph a line? How many should actually be used? Explain.

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

Explain why the \(y\)-values can be any number for the equation \(x=5\). How is this shown in the graph of the equation?

Make Sense? In Exercises 74-77, determine whether each statement makes sense or does not make sense, and explain your reasoning. When finding the slope of the line passing through \((-1,5)\) and \((2,-3)\), I must let \(\left(x_{1}, y_{1}\right)\) be \((-1,5)\) and \(\left(x_{2}, y_{2}\right)\) be \((2,-3)\)

Make Sense? In Exercises 75-78, determine whether each 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 \(\left\\{\left(\frac{8}{11}, \frac{43}{11}\right)\right\\}\), I can use graphs to determine if the solution set is reasonable.

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. A linear function that models tuition and fees at public four-year colleges from 2000 through 2010 has negative slope.

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. Each equation in a system of linear equations has infinitely many ordered-pair solutions.

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. Every system of linear equations has infinitely many ordered-pair solutions.

Make Sense? Determine whether each statement makes sense or does not make sense, and explain your reasoning. I find it easiest to use the addition method when one of the equations has a variable on one side by itself.

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

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Every line in the rectangular coordinate system has an equation that can be expressed in slope-intercept form.

One apartment is directly above a second apartment. The resident living downstairs calls his neighbor living above him and states, "If one of you is willing to come downstairs, we'll have the same number of people in both apartments." The upstairs resident responds, "We're all too tired to move. Why don't one of you come up here? Then we will have twice as many people up here as you've got down there." How many people are in each apartment?

The relationship between Celsius temperature, \(C\), and Fahrenheit temperature, \(F\), can be described by a linear equation in the form \(F=m C+b\). The graph of this equation contains the point \((0,32)\) : Water freezes at \(0^{\circ} \mathrm{C}\) or at \(32^{\circ} \mathrm{F}\). The line also contains the point \((100,212)\) : Water boils at \(100^{\circ} \mathrm{C}\) or at \(212^{\circ} \mathrm{F}\). Write the linear equation expressing Fahrenheit temperature in terms of Celsius temperature.