Chapter 8

In Exercises \(19-24\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 6 r+5 s=3 \\ \frac{3}{2} r-\frac{5}{4} s=\frac{3}{4} \end{array}\right. $$

In Exercises 21-26, solve the system by the method of substitution. $$ \left\\{\begin{array}{c} -5 x+4 y=14 \\ 5 x-4 y=4 \end{array}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{r} -2 x+y=-1 \\ x-2 y=-1 \end{array}\right. $$

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} x-7 y>-36 \\ 5 x+2 y>5 \\ 6 x+5 y>6 \end{array}\right. $$

In Exercises \(19-24\), solve the system by the method of elimination. $$ \left\\{\begin{array}{l} \frac{1}{4} x-y=\frac{1}{2} \\ 4 x+4 y=3 \end{array}\right. $$

In Exercises 21-26, solve the system by the method of substitution. $$ \left\\{\begin{aligned} 3 x-2 y &=3 \\ -6 x+4 y &=-6 \end{aligned}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{l} 2 x+y=-4 \\ 4 x-2 y=8 \end{array}\right. $$

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{lr} y \geq & -6 \\ y \leq & -8 x+9 \\ x \geq & 0 \\ y \leq & 0 \end{array}\right. $$

In Exercises 23-28, use a system of linear equations to determine the number of each type of coin. 35 Nickels and quarters \(\$ 5.75\)

In Exercises 25-28, solve the system by the method of elimination. $$ \left\\{\begin{aligned} -3 x-12 y &=3 \\ 5 x+20 y &=-5 \end{aligned}\right. $$

In Exercises 21-26, solve the system by the method of substitution. $$ \left\\{\begin{array}{rr} -6 x+1.5 y= & 6 \\ 8 x-2 y= & -8 \end{array}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{r} x-2 y=4 \\ 2 x-4 y=8 \end{array}\right. $$

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{r} y \leq 10 \\ x+3 y \leq 15 \\ \geq x \\ y \geq 0 \end{array}\right. $$

In Exercises 23-28, use a system of linear equations to determine the number of each type of coin. 31 Nickels and quarters \(\$ 6.55\)

In Exercises 25-28, solve the system by the method of elimination. $$ \left\\{\begin{array}{r} 7 x+10 y=0 \\ 21 x+30 y=0 \end{array}\right. $$

In Exercises 21-26, solve the system by the method of substitution. $$ \left\\{\begin{array}{r} 0.3 x-0.3 y=0 \\ x-\quad y=4 \end{array}\right. $$

In Exercises 19-26, solve the system by graphing. $$ \left\\{\begin{array}{l} 2 x+3 y=6 \\ 4 x+6 y=12 \end{array}\right. $$

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{array}{rr} x-y & \leq 8 \\ 2 x+5 y & \leq 25 \\ x & \geq 0 \\ y & \geq 0 \end{array}\right. $$

In Exercises 23-28, use a system of linear equations to determine the number of each type of coin. 44 Nickels and dimes \(\$ 3.00\)

In Exercises 25-28, solve the system by the method of elimination. $$ \left\\{\begin{array}{l} 0.4 a+0.7 b=3 \\ 0.8 a+1.4 b=7 \end{array}\right. $$

A total of \(\$ 15,000\) is invested in two funds paying \(5 \%\) and \(8 \%\) simple interest. (There is more risk in the \(8 \%\) fund.) The combined annual interest for the two funds is \(\$ 900\). The system of equations that represents this situation is $$ \left\\{\begin{aligned} x+y &=15,000 \\ 0.05 x+0.08 y &=900 \end{aligned}\right. $$ where \(x\) represents the amount invested in the \(5 \%\) fund and \(y\) represents the amount invested in the \(8 \%\) fund. Solve this system to determine how much of the \(\$ 15,000\) is invested at each rate.

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{aligned} x+7 y &=-5 \\ 3 x-2 y &=8 \end{aligned}\right. $$

In Exercises 23-28, sketch the graph of the system of linear inequalities. $$ \left\\{\begin{aligned} 4 x-y & \leq 13 \\ -x+2 y & \leq 22 \\ x & \geq 0 \\ y & \geq 0 \end{aligned}\right. $$

In Exercises 25-28, solve the system by the method of elimination. $$ \left\\{\begin{array}{r} 0.2 u-0.1 v=1 \\ -0.8 u+0.4 v=3 \end{array}\right. $$

A total of \(\$ 10,000\) is invested in two funds paying \(7 \%\) and \(10 \%\) simple interest. (There is more risk in the \(10 \%\) fund.) The combined annual interest for the two funds is \(\$ 775\). The system of equations that represents this situation is $$ \left\\{\begin{array}{rlr} x+y & =10,000 \\ 0.07 x+0.10 y & =775 \end{array}\right. $$ where \(x\) represents the amount invested in the \(7 \%\) fund and \(y\) represents the amount invested in the \(10 \%\) fund. Solve this system to determine how much of the \(\$ 10,000\) is invested at each rate.

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{aligned} x+2 y &=4 \\ 2 x-2 y &=-1 \end{aligned}\right. $$

In Exercises 29-34, use a system of linear equations to solve the problem. The selling price of an air conditioner is \(\$ 359\). The markup rate is \(30 \%\) of the wholesale cost. Find the wholesale cost.

Ticket sales for a play were \(\$ 3799\) on the first night and \(\$ 4905\) on the second night. On the first night, 213 student tickets and 632 general admission tickets were sold. On the second night, 275 student tickets and 816 general admission tickets were sold. The system of equations that represents this situation is $$ \left\\{\begin{array}{l} 213 x+632 y=3799 \\ 275 x+816 y=4905 \end{array}\right. $$ where \(x\) represents the price of a student ticket and \(y\) represents the price of a general admission ticket. Solve this system to determine the price of each type of ticket.

A solar heating system for a threebedroom home costs \(\$ 28,500\) for installation and \(\$ 125\) per year to operate. An electric heating system for the same home costs \(\$ 5750\) for installation and \(\$ 1000\) per year to operate. The system of equations that represents this situation is \(\begin{cases}y=28,500+125 x & \text { Solar heating } \\ y=5,750+1000 x & \text { Electric heating }\end{cases}\) where \(y\) represents the total cost of heating the home and \(x\) represents the number of years. Solve this system to determine after how many years the total costs for solar heating and electric heating will be the same. What will be the cost at that time?

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{r} -4 x+2 y=12 \\ 2 x+y=6 \end{array}\right. $$

In Exercises 29-34, use a system of linear equations to solve the problem. The sale price of a surround sound system is \(\$ 716\). The discount is \(20 \%\) of the original price. Find the original price.

Ticket sales for an annual variety show were \(\$ 540\) on the first night and \(\$ 850\) on the second night. On the first night, 150 student tickets and 80 general admission tickets were sold. On the second night, 200 student tickets and 150 general admission tickets were sold. The system of equations that represents this situation is $$ \left\\{\begin{array}{l} 150 x+80 y=540 \\ 200 x+150 y=850 \end{array}\right. $$ where \(x\) represents the price of a student ticket and \(y\) represents the price of a general admission ticket. Solve this system to determine the price of each type of ticket.

You are selling tickets for a football game. Student tickets cost \(\$ 3\) each and general admission tickets cost \(\$ 5\) each. You sell 1957 tickets and collect \(\$ 8113\). The system of equations that represents this situation is $$ \left\\{\begin{aligned} x+y &=1957 \\ 3 x+5 y &=8113 \end{aligned}\right. $$ where \(x\) represents the number of students tickets sold and \(y\) represents the number of general admission tickets sold. Solve this system to determine how many of each type of ticket are sold.

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{l} 3 x+2 y=-6 \\ 3 x-2 y=6 \end{array}\right. $$

In Exercises 29-34, use a system of linear equations to solve the problem. The total cost of 8 gallons of regular gasoline and 12 gallons of premium gasoline is \(\$ 75.60\). Premium gasoline costs \(\$ 0.15\) more per gallon than regular gasoline. Find the price per gallon for each type of gasoline.

In your own words, explain the basic steps in solving a system of linear equations by the method of substitution.

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{l} 4 x-5 y=0 \\ 6 x-5 y=10 \end{array}\right. $$

In Exercises 29-34, use a system of linear equations to solve the problem. The total cost of 6 gallons of regular gasoline and 11 gallons of premium gasoline is \(\$ 68.33\). Premium gasoline costs \(\$ 0.20\) more per gallon than regular gasoline. Find the price per gallon for each type of gasoline.

Explain how to solve a system of linear equations by elimination.

When solving a system of linear equations by the method of substitution, how do you recognize that it has no solution?

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{aligned} \frac{1}{2} x+2 y &=-4 \\ -3 x+y &=11 \end{aligned}\right. $$

A person plans to invest up to \(\$ 20,000\) in two different interest-bearing accounts, account \(X\) and account Y. Account \(\mathrm{X}\) is to contain at least \(\$ 5000\). Moreover, account \(\mathrm{Y}\) should have at least twice the amount in account \(X\). Write a system of linear inequalities that describes the various amounts that can be deposited in each account, and sketch the graph of the system.

In Exercises 29-34, use a system of linear equations to solve the problem. A van travels for 2 hours at an average speed of 40 miles per hour. How much longer must the van travel at an average speed of 55 miles per hour so that the average speed for the entire trip will be 45 miles per hour?

When solving a system by the method of elimination, how do you recognize that it has no solution?

When solving a system of linear equations by the method of substitution, how do you recognize that it has infinitely many solutions?

In Exercises 27-36, solve the system by graphing. $$ \left\\{\begin{array}{r} x+\frac{5}{4} y=5 \\ 4 x+5 y=20 \end{array}\right. $$

A person plans to invest up to \(\$ 10,000\) in two different interest-bearing accounts, account \(X\) and account \(Y\). Account \(Y\) is to contain at most \(\$ 3000\). Moreover, account \(X\) should have at least three times the amount in account Y. Write a system of linear inequalities that describes the various amounts that can be deposited in each account, and sketch the graph of the system.

In Exercises 29-34, use a system of linear equations to solve the problem. A van travels for 3 hours at an average speed of 40 miles per hour. How much longer must the van travel at an average speed of 55 miles per hour so that the average speed for the entire trip will be 50 miles per hour?

When solving a system by the method of elimination, how do you recognize that it has infinitely many solutions?

Explain how you can check the solution of a system of linear equations algebraically.