Chapter 11

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{rl}x+y & =10.5 \\ 0.5 x+0.8 y & =7.35\end{array}\right)\)

Solve each system by using either the substitution method or the elimination- by-addition method, whichever seems more appropriate. \(\left(\begin{array}{l}2 x+y=7.75 \\ 3 x+2 y=12.5\end{array}\right)\)

For Problems \(61-80\), solve each problem by using a system of equations. The sum of two numbers is 53 , and their difference is 19. Find the numbers.

For Problems \(61-80\), solve each problem by using a system of equations. The sum of two numbers is \(-3\) and their difference is 25 . Find the numbers.

Solve each problem by using a system of equations. The measure of the larger of two complementary angles is \(15^{\circ}\) more than four times the measure of the smaller angle. Find the measures of both angles.

Solve each problem by using a system of equations. Assume that a plane is flying at a constant speed under unvarying wind conditions. Traveling against a head wind, the plane takes 4 hours to travel 1540 miles. Traveling with a tail wind, the plane flies 1365 miles in 3 hours. Find the speed of the plane and the speed of the wind.

Solve each problem by using a system of equations. The tens digit of a two-digit number is 1 more than three times the units digit. If the sum of the digits is 9 , find the number.

Solve each problem by using a system of equations. The units digit of a two-digit number is 1 less than twice the tens digit. The sum of the digits is 8 . Find the number.

Solve each problem by using a system of equations. The sum of the digits of a two-digit number is 7 . If the digits are reversed, the newly formed number is 9 larger than the original number. Find the original number.

Solve each problem by using a system of equations. The units digit of a two-digit number is 1 less than twice the tens digit. If the digits are reversed, the newly formed number is 27 larger than the original number. Find the original number.

Solve each problem by using a system of equations. A motel rents double rooms at \(\$ 32\) per day and single rooms at \(\$ 26\) per day. If 23 rooms were rented one day for a total of \(\$ 688\), how many rooms of each kind were rented?

Solve each problem by using a system of equations. An apartment complex rents one-bedroom apartments for \$325 per month and two- bedroom apartments for \(\$ 375\) per month. One month the number of onebedroom apartments rented was twice the number of two-bedroom apartments. If the total income for that month was \(\$ 12,300\), how many apartments of each kind were rented?

Solve each problem by using a system of equations. The income from a student production was \(\$ 10,000\). The price of a student ticket was \(\$ 3\), and nonstudenttickets were sold at \(\$ 5\) each. Three thousand tickets were sold. How many tickets of each kind were sold?

Solve each problem by using a system of equations. Michelle can enter a small business as a full partner and receive a salary of \(\$ 10,000\) a year and \(15 \%\) of the year's profit, or she can be sales manager for a salary of \(\$ 25,000\) plus \(5 \%\) of the year's profit. What must the year's profit be for her total earnings to be the same whether she is a full partner or a sales manager?

Solve each problem by using a system of equations. Melinda invested three times as much money at \(11 \%\) yearly interest as she did at \(9 \%\). Her total yearly interest from the two investments was \(\$ 210\). How much did she invest at each rate?

Solve each problem by using a system of equations. One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 20 miles in 4 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 19 miles in 1 hour. Find the rate of the current.

Solve each problem by using a system of equations. One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 20 miles in 4 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 19 miles in 1 hour. Find the rate of the current.

Solve each problem by using a system of equations. One solution contains \(30 \%\) alcohol and a second solution contains \(70 \%\) alcohol. How many liters of each solution should be mixed to make 10 liters containing \(40 \%\) alcohol?

Solve each problem by using a system of equations. Bill bought 4 tennis balls and 3 golf balls for a total of \(\$ 10.25\). Bret went into the same store and bought 2 tennis balls and 5 golf balls for \(\$ 11.25\). What was the price for a tennis ball and the price for a golf ball?

Solve each problem by using a system of equations. Six cans of pop and 2 bags of potato chips cost \(\$ 5.12\). At the same prices, 8 cans of pop and 5 bags of potato chips cost \(\$ 9.86\). Find the price per can of pop and the price per bag of potato chips.

Solve each problem by using a system of equations. A cash drawer contains only five- and ten-dollar bills. There are 12 more five-dollar bills than ten-dollar bills. If the drawer contains \(\$ 330\), find the number of each kind of bill.

Solve each problem by using a system of equations. Brad has a collection of dimes and quarters totaling \(\$ 47.50\). The number of quarters is 10 more than twice the number of dimes. How many coins of each kind does he have?

Give a general description of how to use the substitution method to solve a system of two linear equations in two variables.

Give a general description of how to use the elimination-by-addition method to solve a system of two linear equations in two variables.

Which method would you use to solve the system \(\left(\begin{array}{l}9 x+4 y=7 \\ 3 x+2 y=6\end{array}\right) ?\) Why?

Which method would you use to solve the system \(\left(\begin{array}{l}5 x+3 y=12 \\ 3 x-y=10\end{array}\right) ?\) Why?

For Problems \(85-90\), solve each system. \(\left(\begin{array}{l}\frac{1}{x}+\frac{2}{y}=\frac{7}{12} \\\ \frac{3}{x}-\frac{2}{y}=\frac{5}{12}\end{array}\right)\)

Solve each system. \(\left(\begin{array}{l}\frac{3}{x}+\frac{2}{y}=2 \\\ \frac{2}{x}-\frac{3}{y}=\frac{1}{4}\end{array}\right)\)

Solve each system. \(\left(\begin{array}{l}\frac{3}{x}-\frac{2}{y}=\frac{13}{6} \\\ \frac{2}{x}+\frac{3}{y}=0\end{array}\right)\)

Solve each system. \(\left(\begin{array}{l}\frac{4}{x}+\frac{1}{y}=11 \\\ \frac{3}{x}-\frac{5}{y}=-9\end{array}\right)\)

Solve each system. \(\left(\begin{array}{l}\frac{5}{x}-\frac{2}{y}=23 \\\ \frac{4}{x}+\frac{3}{y}=\frac{23}{2}\end{array}\right)\)

Solve each system. \(\left(\begin{array}{l}\frac{2}{x}-\frac{7}{y}=\frac{9}{10} \\\ \frac{5}{x}+\frac{4}{y}=-\frac{41}{20}\end{array}\right)\)

Consider the linear system \(\left(\begin{array}{l}a_{1} x+b_{1} y=c_{1} \\\ a_{2} x+b_{2} y=c_{2}\end{array}\right)\). (a) Prove that this system has exactly one solution if and only if \(\frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}}\). (b) Prove that this system has no solution if and only if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} .\) (c) Prove that this system has infinitely many solutions if and only if \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\).

For each of the systems of equations in Problem 92, use your graphing calculator to help determine whether the system is consistent or inconsistent or whether the equations are dependent. (a) \(\left(\begin{array}{l}y=3 x-1 \\ y=9-2 x\end{array}\right)\) (b) \(\left(\begin{array}{rl}5 x+y & =-9 \\ 3 x-2 y & =5\end{array}\right)\) (c) \(\left(\begin{array}{l}4 x-3 y=18 \\ 5 x+6 y=3\end{array}\right)\) (d) \(\left(\begin{array}{l}2 x-y=20 \\ 7 x+y=79\end{array}\right)\) (e) \(\left(\begin{array}{l}13 x-12 y=37 \\ 15 x+13 y=-11\end{array}\right)\) (f) \(\left(\begin{array}{l}1.98 x+2.49 y=13.92 \\ 1.19 x+3.45 y=16.18\end{array}\right)\)

Use your graphing calculator to help determine the solution set for each of the following systems. Be sure to check your answers. (a) \(\left(\begin{array}{l}y=3 x-1 \\ y=9-2 x\end{array}\right)\) (b) \(\left(\begin{array}{rl}5 x+y & =-9 \\ 3 x-2 y & =5\end{array}\right)\) (c) \(\left(\begin{array}{rl}4 x-3 y & =18 \\ 5 x+6 y & =3\end{array}\right)\) (d) \(\left(\begin{array}{l}2 x-y=20 \\ 7 x+y=79\end{array}\right)\) (e) \(\left(\begin{array}{l}13 x-12 y=37 \\ 15 x+13 y=-11\end{array}\right)\) (f) \(\left(\begin{array}{l}1.98 x+2.49 y=13.92 \\ 1.19 x+3.45 y=16.18\end{array}\right)\)