Chapter 4

Involve dual investments. You invest 6000 dollar in two accounts paying \(6 \%\) and \(9 \%\) annual interest. At the end of the year, the accounts earn the same interest. How much was invested at each rate?

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{aligned} 7 x-3 y &=4 \\ -14 x+6 y &=-7 \end{aligned}\right.$$

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x+y=5 \\ 2 x+2 y=12\end{array}\right.$$

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The sum of two numbers is \(81 .\) One number is 41 more than the other. Find the numbers.

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}0 \leq x \leq 3 \\\0 \leq y \leq 3\end{array}\right.$$

Involve dual investments. You invest 7200 dollar in two accounts paying \(8 \%\) and \(10 \%\) annual interest. At the end of the year, the accounts earn the same interest. How much was invested at each rate?

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 2 x+4 y=5 \\ 3 x+6 y=6 \end{array}\right.$$

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x-y=2 \\ 3 x-3 y=-6\end{array}\right.$$

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The sum of two numbers is \(62 .\) One number is 12 more than the other. Find the numbers.

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}0 \leq x \leq 5 \\\0 \leq y \leq 5\end{array}\right.$$

Involve dual investments. Your grandmother needs your help. She has 50,000 dollar to invest. Part of this money is to be invested in noninsured bonds paying \(15 \%\) annual interest. The rest of this money is to be invested in a government-insured certificate of deposit paying \(7 \%\) annual interest. She told you that she requires a total of 6000 dollar per year in extra income from these investments. How much money should be placed in each investment?

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 5 x+y=2 \\ 3 x+y=1 \end{array}\right.$$

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x-y=0 \\ y=x\end{array}\right.$$

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The difference between two numbers is \(5 .\) Four times the larger number is 6 times the smaller number. Find the numbers.

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}x-y \leq 4 \\\x+2 y \leq 4\end{array}\right.$$

Involve dual investments. Things did not go quite as planned. You invested part of 8000 dollar in an account that paid \(12 \%\) annual interest. However, the rest of the money suffered a \(5 \%\) loss. If the total annual income from both investments was 620 dollar how much was invested at each rate?

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 2 x-5 y=-1 \\ 2 x-y=1 \end{array}\right.$$

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The difference between two numbers is \(25 .\) Two times the larger number is 12 times the smaller number. Find the numbers.

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}2 x-y=0 \\ y=2 x\end{array}\right.$$

Graph the solution set of each system of linear inequalities. $$\left\\{\begin{array}{l}x-y \leq 3 \\\2 x+y \leq 4\end{array}\right.$$

Involve mixtures A lab technician needs to mix a \(5 \%\) fungicide solution with a \(10 \%\) fungicide solution to obtain a 50 -liter mixture consisting of \(8 \%\) fungicide. How many liters of each of the fungicide solutions must be used? Begin by filling in the missing entries in the table on the next page. Then use the fact that the amount of fungicide in the \(5 \%\) solution plus the amount of fungicide in the \(10 \%\) solution must equal the amount of fungicide in the \(8 \%\) mixture. (TABLE CAN NOT COPY)

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} x=5-3 y \\ 2 x+6 y=10 \end{array}\right.$$

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x=2 \\ y=4\end{array}\right.$$

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The difference between two numbers is 1. The sum of the larger number and twice the smaller number is 7. Find the numbers.

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}x+y \geq 1 \\\x-y \geq 1 \\\x \geq 4\end{array}\right.$$

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 4 x=36+8 y \\ 3 x-6 y=27 \end{array}\right.$$

Write a system of equations modeling the given conditions. Then solve the system by the substitution method and find the two numbers. The difference between two numbers is \(5 .\) The sum of the larger number and twice the smaller number is \(14 .\) Find the numbers.

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x=3 \\ y=5\end{array}\right.$$

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}x-y \leq 3 \\\x+y \leq 3 \\\x \geq-2\end{array}\right.$$

Involve mixtures How many ounces of a \(15 \%\) alcohol solution must be mixed with 4 ounces of a \(20 \%\) alcohol solution to make a \(17 \%\) alcohol solution?

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 4(3 x-y)=0 \\ 3(x+3)=10 y \end{array}\right.$$

Multiply each equation in the system by an appropriate number so that the coefficients are integers. Then solve the system by the substitution method. $$\left\\{\begin{array}{l}0.7 x-0.1 y=0.6 \\ 0.8 x-0.3 y=-0.8\end{array}\right.$$

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x=2 \\ x=-1\end{array}\right.$$

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}x+2 y<6 \\\y>2 x-2 \\\y \geq 2\end{array}\right.$$

Involve mixtures How many ounces of a \(50 \%\) alcohol solution must be mixed with 80 ounces of a \(20 \%\) alcohol solution to make a \(40 \%\) alcohol solution?

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} 2(2 x+3 y)=0 \\ 7 x=3(2 y+3)+2 \end{array}\right.$$

Multiply each equation in the system by an appropriate number so that the coefficients are integers. Then solve the system by the substitution method. $$\left\\{\begin{array}{l}1.25 x-0.01 y=4.5 \\ 0.5 x-0.02 y=1\end{array}\right.$$

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}x=3 \\ x=-2\end{array}\right.$$

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}2 x-3 y<6 \\\2 x-3 y>-6 \\\\-3 \leq x \leq 2\end{array}\right.$$

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} x+y=11 \\ \frac{x}{5}+\frac{y}{7}=1 \end{array}\right.$$

b. Use your answer from part (a) to complete this statement: If workers are paid million available workers and \(-\) million workers \(-1\) will be hired. c. In 2007 , the federal minimum wage was set at \(\$ 5.15\) per hour. Substitute 5.15 for \(p\) in the demand model, \(p=-0.325 x+5.8,\) and determine the millions of workers employers will hire at this price. d. At a minimum wage of \(\$ 5.15\) per hour, use the supply model, \(p=0.375 x+3,\) to determine the millions of available workers. Round to one decimal place. e. At a minimum wage of \(\$ 5.15\) per hour, use your answers from parts (c) and (d) to determine how many more people are looking for work than employers are willing to hire.

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}y=0 \\ y=4\end{array}\right.$$

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}y \leq-3 x+3 \\\y \geq-x-1 \\\y

Involve mixtures At the north campus of a small liberal arts college, \(10 \%\) of the students are women. At the south campus, \(50 \%\) of the students are women. The campuses are merged into one east campus. If \(40 \%\) of the 1200 students at the east campus are women, how many students did each of the north and south campuses have before the merger?

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} x-y=-3 \\ \frac{x}{9}-\frac{y}{7}=-1 \end{array}\right.$$

Solve each system by graphing. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l}y=0 \\ y=5\end{array}\right.$$

Graph the solution set of each system of linear inequalities. If the system has no solutions, state this and explain why. $$\left\\{\begin{array}{l}y>-3 x+5 \\\y \geq-x+3 \\\y \geq \frac{1}{2} x \\\x \geq 0 \\\y \geq 0\end{array}\right.$$

Describe the conditions in a problem that enable it to be solved using a system of linear equations.

In Exercises \(1-44,\) solve each system by the addition method. If there is no solution or an infinite number of solutions, so state. Use set notation to express solution sets. $$\left\\{\begin{array}{l} \frac{4}{5} x-y=-1 \\ \frac{2}{5} x+y=1 \end{array}\right.$$

Describe a problem that might arise when solving a system of equations using graphing. Assume that both equations in the system have been graphed correctly and the system has exactly one solution.