Chapter 6

A potter has fixed costs of \(\$ 80 .\) It costs her \(\$ 12\) to produce each piece, and she sells each piece for \(\$ 20 .\) Therefore, her total cost \(C\) for producing \(n\) pieces is given by the equation \(C=80+12 n .\) Her total revenue \(R\) for producing \(n\) pieces is given by the equation \(R=20 n\). (A) Sketch the graph of both equations on the same coordinate system, labeling the horizontal axis \(n\) (B) The potter will break even when her costs and revenue are equal. Use the graph in part (a) to determine the point at which the two lines cross. This is called the break-even point. (C) How many pieces must she sell to break even?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} x+y=1 \\ x-y=3 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a oneor a two-variable approach. The sum of two numbers is 130. If their difference is 28, find the two numbers.

A small software firm has a new computer game that it wants to market on a CD- ROM. The fixed costs are \(\$ 2400\), and it costs \(\$ 6\) to produce each CD- ROM. Therefore, their total cost \(C\) for producing \(n\) CDs is given by the equation \(C=2400+6 n .\) They plan on selling the CDs for \(\$ 18\) each. Their total revenue \(R\) for producing \(n\) CDs is given by the equation \(R=18 n\). (A). Sketch the graph of both equations on the same coordinate system, labeling herizontal axis \(n\) and the vertical axis \(C\). (B) The company will break even when their costs and revenue are equal. Use the graph in part (a) to determine the point at which the two lines cross. This is called the break-even point. (C) How many CDs must they sell to break even?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} x-y=4 \\ x+y=6 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The difference between two numbers is \(3 .\) If the sum of twice the larger and the smaller is \(48,\) find the numbers.

The Metropolitan Transportation Authority charges \(\$ 1.75\) per ride on public transportation. They offer a monthly commuter pass for \(\$ 48\) that allows unlimited travel on the public transportation system. Let \(n\) represent the number of trips taken per month on public transportation and \(C\) represent the cost of all these trips. (A) Write an equation for the transportation cost \(C\) if you buy the monthly pass and if you pay for each trip individually. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(n\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine how many trips per month make it more economical to buy a monthly pass rather than pay per trip.

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{r} 2 x+y=5 \\ x-y=4 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Sam has 80 coins consisting of nickels and quarters. If the total value of the coins is \(\$ 13.60,\) how many of each type of coin are there?

Cellmate Communications offers two monthly cellular phone plans. The Standard plan costs \(\$ 15\) per month plus \(\$ 0.22\) per minute of air time. The Deluxe plan costs \(\$ 35\) per month plus \(\$ 0.14\) per minute of air time. (A) Write an equation for the monthly cost \(C\) of the Standard plan and the Deluxe plan for a month in which you use \(m\) minutes. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(m\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine how many air time minutes per month make it more economical to buy the Standard plan.

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{aligned} -x+3 y &=8 \\ x-2 y &=-6 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Susan has 92 packages in her truck. Some of the packages weigh 32 lb each, and the rest weigh 12 lb each. If the total weight of all the packages is 1604 lb, how many of the lighter packages are there on the truck?

The Lease-From-Us Company offers two different leasing plans for their top- ofthe-line color copying machine. The Economy plan costs \(\$ 175\) per month plus \(\$ 0.032\) per copy. The Standard plan costs \(\$ 225\) per month plus \(\$ 0.024\) per copy. (A) Write an equation for the monthly cost \(C\) of the Standard plan and the Economy for a month in which you make \(n\) copies. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(n\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine how many copies per month make it more economical to buy the Economy plan.

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} 7 x+2 y-15=0 \\ 3 x-2 y+5=0 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Two cars start at the same place and time, and travel in opposite directions. One car is traveling 15 kph faster than the other. After 5 hours the two cars are \(275 \mathrm{km}\) apart. Find the speed of each car.

An appliance store offers two lease plans for renting a refrigerator. Plan A costs a Elat fee of \(\$ 75\) plus a monthly rental fee of \(\$ 28 .\) Plan \(B\) costs a flat fee of \(\$ 50\) Slus a monthly rental fee of \(\$ 34\). (A) Write an equation for the total cost \(C\) of renting a refrigerator under both plans for \(m\) months. (B) Sketch the graphs of the two equations obtained in part (a). Label the horizontal axis \(m\) and the vertical axis \(C\). (C) Using the graphs obtained in part (b), determine when each plan is more economical.

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} a-5 b-30=0 \\ a+5 b+40=0 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The ratio of two positive numbers is 3 to \(4 .\) If one of the numbers is 5 more than the other, what are the two numbers?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{aligned} 2 x+y &=15 \\ x-2 y &=0 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Carmen invests a total of \(\$ 1,700\) in two stocks. One stock pays a yearly dividend of \(7 \%,\) while the other pays \(6 \% .\) If Carmen received \(\$ 110\) in combined dividends from the two stocks, how much did she invest in each?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{r} x-3 y=1 \\ 2 x+y=9 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A storekeeper is preparing a mixture of peanuts and raisins. If peanuts cost \(\$ 1.60\) per pound and raisins cost \(\$ 1.85\) per pound, how many pounds of each should be used to prepare 50 pounds of a mixture selling at \(\$ 1.70\) per pound?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$$$\left\\{\begin{array}{c} 3 x+2 y=-11 \\ x+3 y=1 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Tim and Marge go into a music shop. Tim buys four CDs and six DVDs for a total of \(\$ 165.50\), while Marge buys five CDs and 3 DVDs for a total of \(\$ 121.60\). What are the prices of an individual CD and an individual DVD?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{aligned} 5 x-y &=18 \\ x+2 y &=-3 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A discount building supplies store sells both first-quality and second-quality floor tiles. Robin buys three cases of first-quality tiles and one case of second-quality tiles for a total of \(\$ 66,\) while Gene buys one case of first-quality tiles and three cases of second-quality tiles for a total of \(\$ 54 .\) What is the cost per case for each type of tile?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} 4 s-5 t=4 \\ 2 s+10 t=7 \end{array}\right.$$

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{c}3 x+y=6 \\ 6 x+2 y=12\end{array}\right.$$.

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The length of a rectangle is twice its width. If the perimeter of the rectangle is 28 in., what are its dimensions?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} 6 u-w=2 \\ 2 u-3 w=2 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. How many \(37c\) and \(80c\) stamps did Joe buy if he bought 50 stamps and paid \(\$ 24.52\) for them?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} 12 c-20 d=19 \\ 18 c-12 d=15 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Pat and Carlos both belong to the same book club. Pat orders two regular selections and three specially discounted ones for a total of \(\$ 56.90 .\) Carlos orders three regular selections and four specially discounted ones for a total of \(\$ 80.85\) What are the prices of a regular and a specially discounted selection?

$$\text { In Exercises } 1-14, \text { solve the system of equations using the elimination method.}$$ $$\left\\{\begin{array}{l} 14 w+9 z=6 \\ 21 w+15 z=11 \end{array}\right.$$

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{r}2 x-y=0 \\ x-2 y=0\end{array}\right.$$.

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. On Monday John walks for 1 hour, jogs for 2 hours, and covers a total of \(28 \mathrm{km}\). On Tuesday he walks for 2 hours, jogs for 1 hour, and covers a total of \(23 \mathrm{km} .\) What are his rate walking and his rate jogging?

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} x+2 y &=9 \\ y &=3 x+1 \end{aligned}\right.$$

Sketch a graph that represents the scenario described in the exercise. Be sure to clearly label any variables and the coordinate axes. Keep in mind that various graphs may be drawn to represent each situation. Suppose that the temperature of a metal bar starts at \(40^{\circ} \mathrm{C}\). Over the course of 30 minutes the temperature of the bar steadily increases to a temperature of \(160^{\circ} \mathrm{C} .\) It remains at that temperature for 45 minutes and then steadily cools off to a temperature of \(60^{\circ} \mathrm{C}\) over the next 20 minutes.

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The ratio of two positive numbers is 6 to \(5 .\) If the difference between the two numbers is \(8,\) what are the numbers?

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{c}5 x+y=10 \\ x+y=6\end{array}\right.$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} x+3 y &=5 \\ y &=4 x-7 \end{aligned}\right.$$

Sketch a graph that represents the scenario described in the exercise. Be sure to clearly label any variables and the coordinate axes. Keep in mind that various graphs may be drawn to represent each situation. The price of a certain stock starts the day at \(\$ 15\) per share. Over the first 2 hours of trading, the price of the stock steadily declines to \(\$ 13\) per share. It remains at that price for 3 hours and then declines to \(\$ 11.50\) per share over the next hour.

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Seven thousand tickets worth \(\$ 137,125\) were sold for a concert. General admission tickets cost \(\$ 22\) each, and standing-room-only tickets cost \(\$ 14.50\) each. How many of each type were sold?

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{l}3 x-y=9 \\ 6 x-2 y=18\end{array}\right.$$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} 2 x-3 y &=8 \\ x &=2 y+4 \end{aligned}\right.$$

Simplify: $$\frac{6 x^{2}}{25 y z^{3}} \div(15 x y z)$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Two airplanes leave an airport at the same time, flying in opposite directions. One plane is flying at twice the speed of the other. If after 4 hours they are \(1,800\) miles apart, find the speed of each plane.

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{l}8 x-6 y=24 \\ 4 x-3 y=12\end{array}\right.$$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} 3 x+4 y &=12 \\ x &=3 y-9 \end{aligned}\right.$$

Solve for \(t:\) $$\frac{t+3}{5}-\frac{t-2}{4} \leq 2$$