Chapter 6

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{l}5 x-3 y=15 \\ 2 x-y=4\end{array}\right.$$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} 5 x+4 y &=6 \\ x-y &=3 \end{aligned}\right.$$

A total of \(\$ 1185\) was spent on 36 pairs of slacks. Some were dress slacks costing \(\$ 40\) each, and the rest were casual slacks costing \(\$ 25\) each. How many of each were bought?

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Two retailers are ordering from the same source. One retailer orders eight stereo receivers and four turntables at a total cost of \(\$ 2060 .\) A second retailer orders five of the same receivers and six of the same turntables at a total cost of \(\$ 1690\). What are the costs of an individual receiver and an individual turntable?

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{array}{c} 3 a+5 b=1 \\ b-a=5 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A \(25 \%\) iodine solution is to be mixed with a \(75 \%\) iodine solution to produce 5 gallons of a \(70 \%\) iodine solution. How many gallons of each solution are needed?

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{l}y=2 x+6 \\ y=x+1\end{array}\right.$$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} 2 r-5 s &=9 \\ s &=1-r \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. John goes into a donut shop and buys ten plain donuts and five cream-filled donuts for \(\$ 3.70 .\) Jane goes into the same shop and buys five plain donuts and ten cream-filled donuts for \(\$ 4.10 .\) What is the cost of a plain donut?

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{l}y=3 x-9 \\ y=x-5\end{array}\right.$$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{aligned} 4 u+3 v &=0 \\ u &=-1-v \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. In a recent school election, 571 votes were cast for class president. If the winner received 89 more votes than the loser, how many votes did each receive?

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{array}{l} 7 x+2 y=9 \\ 2 x+3 y=5 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A bookstore buys 35 books for \(\$ 271 .\) Some of the books cost \(\$ 7\) each and the remainder cost \(\$ 9\) each. How many of each type were bought?

Use the graphical method to solve the given system of equations for \(x\) and \(y .\) $$\left\\{\begin{array}{l}y=x+5 \\\y=x-2\end{array}\right.$$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{array}{l} 6 x-5 y=2 \\ 5 x-2 y=6 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. An artist's supply store sold a total of 20 canvases for \(\$ 172\). If some of the canvases cost \(\$ 7.50\) each and the remainder cost \(\$ 10.25\) each, how many of each type were sold?

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{array}{l} 6 u-w=2 \\ 2 u-3 w=2 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Two cars start traveling directly toward each other at the same time from positions \(480 \mathrm{km}\) apart. They meet after 4 hours. If one car travels \(40 \mathrm{kph}\) faster than the other car, find the speed of each car.

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

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{array}{r} a-4 b=1 \\ 2 a-5 b=3 \end{array}\right.$$

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{array}{l} 4 w-3 t=8 \\ 6 w-t=5 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A small single-engine plane travels 150 miles per hour with a tailwind and 90 miles per hour with a headwind. Find the speed of the wind and the speed of the plane in still air.

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

$$\text { In Exercises } 15-28, \text { solve the system of equations using the substitution method.}$$ $$\left\\{\begin{array}{c} 18 p+2 r=1 \\ 6 p-r=2 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A \(60 \%\) acid solution is to be mixed with an \(80 \%\) acid solution to produce 20 liters of a \(65 \%\) acid solution. How many liters of each solution are needed?

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{c} r+2 t=10 \\ 3 r+t=-15 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A coffee wholesaler wishes to produce 60 pounds of a coffee blend selling at \(\$ 3.10\) per pound. How many pounds of coffee blends selling at \(\$ 3.35\) per pound and \(\$ 2.75\) per pound should be mixed to produce such a mixture?

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 5 m+n &=5 \\ m &=2 n+12 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Mrs. Thomas has \(\$ 15,000\) to invest. She will invest part of it in a corporate bond that pays \(8 \%\) interest per year and the rest in a stock that pays \(11 \%\) interest per year. How should she divide up the \(\$ 15,000\) so that her total yearly interest will be \(10 \%\) of her investments?

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{l} 6 x+y=6 \\ 4 x+1=y \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A company is trying to determine which computer system to install. System A consists of a central minicomputer costing \(\$ 100,000\) plus desktop terminals costing \(\$ 800\) each. System \(B\) is a network of desktop personal computers that costs \(\$ 16,000\) to install plus \(\$ 1,200\) for each desktop personal computer. How many desktop setups would the company need in order to make the costs of the two systems equal?

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 3 x+6 y &=2 \\ -3 x-3 y &=1 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. Lenore can purchase a car for \(\$ 15,000,\) which will require her to spend an average of \(\$ 80\) per month in repairs and maintenance, or she can lease a car for \(\$ 350\) per month, which includes all repairs and maintenance. After how many months will the leased car and the purchased car cost the same?

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 8 a+6 b &=-3 \\ 12 a+9 b &=-5 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A mathematics department has budgeted \(\$ 10,000\) to purchase computers and printers. On this fixed budget they can purchase 10 computers and 10 printers, or they can purchase 12 computers and 2 printers. Find the individual costs of a computer and printer.

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{array}{r} 8 a+6 b=6 \\ 12 a-9 b=3 \end{array}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A small company has budgeted \(\$ 6,000\) per month to lease vehicles. On this budget, the company can lease 12 cars and 4 trucks each month, or 8 cars and 6 trucks. Find the monthly cost to lease a car and to lease a truck.

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 11 a-2 b &=30 \\ 3 a+3 b &=-6 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The perimeter of a rectangle is \(46 \mathrm{cm} .\) Twice the width is 1 more than the length. Find the dimensions of the rectangle.

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$ \left\\{\begin{array}{l} 7 a-5 b=17 \\ 3 a-2 b=7 \end{array}\right. $$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The perimeter of a rectangle is \(96 \mathrm{ft}\). Three times the width is 8 less than the length. Find the dimensions of the rectangle.

Use the graphical method to solve the given system of equations for \(x\) and \(y.\) $$\left\\{\begin{aligned} 2 x-y &=2 \\ x &=-4 \end{aligned}\right.$$

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 2 x+3 &=4 y \\ 6 x &=9-12 y \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A bag contains 36 marbles, some of which are red and the remainder of which are blue. Twice the number of red marbles is six less than the number of blue marbles. Find the number of marbles of each color.

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 3 y &=24-9 x \\ 3 x+y &=8 \end{aligned}\right.$$

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. A certain mutual fund contains 100 stocks. On a certain day all of the stocks changed price. Three times the number of stocks that went up is 14 more than 8 times the number of stocks that went down. Find how many stocks went up and how many went down.

Use the graphical method to solve the given system of equations for \(x\) and \(y.\) $$\left\\{\begin{array}{l}y=4 \\ x=-1\end{array}\right.$$

In Exercises \(29-62,\) solve the system of equations using any method you choose. $$\left\\{\begin{aligned} 5 x+2 y &=4 y+9 \\ y &=x-3 \end{aligned}\right.$$

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