Chapter 5

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 7 x-9 y=16 \\ -21 x+27 y=-24 \end{array}\right. $$

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 5 x-3 y=15 \\ y=\frac{5}{3} x-2 \end{array}\right. $$

In the following exercises, solve the systems of equations by substitution. $$ \left\\{\begin{array}{l} 2 x+4 y=7 \\ y=-\frac{1}{2} x-4 \end{array}\right. $$

After four years in college, Josie owes \(\$ 65,800\) in student loans. The interest rate on the federal loans is \(4.5 \%\) and the rate on the private bank loans is \(2 \%\). The total interest she owed for one year was \(\$ 2,878.50 .\) What is the amount of each loan?

The sum of two numbers is \(65 .\) Their difference is \(25 .\) Find the numbers.

The sum of two numbers is 37. Their difference is 9 . Find the numbers.

The sum of two numbers is -27. Their difference is -59. Find the numbers.

The sum of two numbers is -45. Their difference is -89. Find the numbers.

Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for \(\$ 114\) or she is able to buy 2 shirts and 4 sweaters for \(\$ 164 .\) How much does a shirt cost? How much does a sweater cost?

Peter is buying office supplies. He is able to buy 3 packages of paper and 4 staplers for \(\$ 40\) or he is able to buy 5 packages of paper and 6 staplers for \(\$ 62\). How much does a package of paper cost? How much does a stapler cost?

The total amount of sodium in 2 hot dogs and 3 cups of cottage cheese is \(4720 \mathrm{mg}\). The total amount of sodium in 5 hot dogs and 2 cups of cottage cheese is \(6300 \mathrm{mg}\). How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?

The total number of calories in 2 hot dogs and 3 cups of cottage cheese is 960 calories. The total number of calories in 5 hot dogs and 2 cups of cottage cheese is 1190 calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination. (a) \(\left\\{\begin{array}{l}8 x-15 y=-32 \\ 6 x+3 y=-5\end{array}\right.\) (B) \(\left\\{\begin{array}{l}x=4 y-3 \\ 4 x-2 y=-6\end{array}\right.\)

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination. (a) \(\left\\{\begin{array}{l}y=7 x-5 \\ 3 x-2 y=16\end{array}\right.\) (b) \(\left\\{\begin{array}{l}12 x-5 y=-42 \\ 3 x+7 y=-15\end{array}\right.\)

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination. (a) \(\quad\left\\{\begin{array}{l}y=4 x+9 \\ 5 x-2 y=-21\end{array}\right.\) (b) \(\left\\{\begin{array}{l}9 x-4 y=24 \\ 3 x+5 y=-14\end{array}\right.\)

In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination. (a) \(\left\\{\begin{array}{l}14 x-15 y=-30 \\ 7 x+2 y=10\end{array}\right.\) (b) \(\left\\{\begin{array}{l}x=9 y-11 \\ 2 x-7 y=-27\end{array}\right.\)

179\. Norris can row 3 miles upstream against the current in 1 hour, the same amount of time it takes him to row 5 miles downstream, with the current. Solve the system. \(\left\\{\begin{array}{l}r-c=3 \\ r+c=5\end{array}\right.\) (a) for \(r,\) his rowing speed in still water. (b) Then solve for \(c,\) the speed of the river current.

Josie wants to make 10 pounds of trail mix using nuts and raisins, and she wants the total cost of the trail mix to be \(\$ 54 .\) Nuts cost \(\$ 6\) per pound and raisins cost \(\$ 3\) per pound. Solve the system \(\left\\{\begin{array}{l}n+r=10 \\ 6 n+3 r=54\end{array}\right.\) to find \(n,\) the number of pounds of nuts, and \(r,\) the number of pounds of raisins she should use.

Solve the system \(\left\\{\begin{array}{l}x+y=10 \\ 5 x+8 y=56\end{array}\right.\) (a) by substitution (b) by graphing \(\odot\) Which method do you prefer? Why?

Solve the system \(\left\\{\begin{array}{l}x+y=-12 \\ y=4-\frac{1}{2} x\end{array}\right.\) (a) by substitution (b) by graphing (c) Which method do you prefer? Why?

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is fifteen. One number is three less than the other. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is twenty-five. One number is five less than the other. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is negative thirty. One number is five times the other. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. The sum of two numbers is negative sixteen. One number is seven times the other. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. Twice a number plus three times a second number is twentytwo. Three times the first number plus four times the second is thirty-one. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. Six times a number plus twice a second number is four. Twice the first number plus four times the second number is eighteen. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. Three times a number plus three times a second number is fifteen. Four times the first plus twice the second number is fourteen. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. Three times a number plus three times a second number is fifteen. Four times the first plus twice the second number is fourteen. Find the numbers.

In the following exercises, translate to a system of equations and solve the system. A married couple together earn \(\$ 75,000\). The husband earns \(\$ 15,000\) more than five times what his wife earns. What does the wife earn?

In the following exercises, translate to a system of equations and solve the system. During two years in college, a student earned \(\$ 9,500\). The second year she earned \(\$ 500\) more than twice the amount she earned the first year. How much did she earn the first year?

In the following exercises, translate to a system of equations and solve the system. Daniela invested a total of \(\$ 50,000,\) some in a certificate of deposit (CD) and the remainder in bonds. The amount invested in bonds was \(\$ 5000\) more than twice the amount she put into the CD. How much did she invest in each account?

In the following exercises, translate to a system of equations and solve the system. Jorge invested \(\$ 28,000\) into two accounts. The amount he put in his money market account was \(\$ 2,000\) less than twice what he put into a CD. How much did he invest in each account?

In the following exercises, translate to a system of equations and solve the system. In her last two years in college, Marlene received \(\$ 42,000\) in loans. The first year she received a loan that was \(\$ 6,000\) less than three times the amount of the second year's loan. What was the amount of her loan for each year?

In the following exercises, translate to a system of equations and solve the system. Jen and David owe \(\$ 22,000\) in loans for their two cars. The amount of the loan for Jen's car is \(\$ 2000\) less than twice the amount of the loan for David's car. How much is each car loan?

Alyssa is twelve years older than her sister, Bethany. The sum of their ages is forty-four. Find their ages.

Robert is 15 years older than his sister, Helen. The sum of their ages is sixty-three. Find their ages.

The age of Noelle's dad is six less than three times Noelle's age. The sum of their ages is seventyfour. Find their ages.

The age of Mark's dad is 4 less than twice Marks's age. The sum of their ages is ninety-five. Find their ages.

Two containers of gasoline hold a total of fifty gallons. The big container can hold ten gallons less than twice the small container. How many gallons does each container hold?

June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?

Shelly spent 10 minutes jogging and 20 minutes cycling and burned 300 calories. The next day, Shelly swapped times, doing 20 minutes of jogging and 10 minutes of cycling and burned the same number of calories. How many calories were burned for each minute of jogging and how many for each minute of cycling?

Drew burned 1800 calories Friday playing one hour of basketball and canoeing for two hours. Saturday he spent two hours playing basketball and three hours canoeing and burned 3200 calories. How many calories did he burn per hour when playing basketball?

Troy and Lisa were shopping for school supplies. Each purchased different quantities of the same notebook and thumb drive. Troy bought four notebooks and five thumb drives for \(\$ 116\). Lisa bought two notebooks and three thumb dives for \(\$ 68\). Find the cost of each notebook and each thumb drive.

Nancy bought seven pounds of oranges and three pounds of bananas for \(\$ 17\). Her husband later bought three pounds of oranges and six pounds of bananas for \(\$ 12 .\) What was the cost per pound of the oranges and the bananas?

In the following exercises, translate to a system of equations and solve. The difference of two complementary angles is 30 degrees. Find the measures of the angles.

In the following exercises, translate to a system of equations and solve. The difference of two complementary angles is 68 degrees. Find the measures of the angles.

In the following exercises, translate to a system of equations and solve. The difference of two supplementary angles is 70 degrees. Find the measures of the angles.

In the following exercises, translate to a system of equations and solve. The difference of two supplementary angles is 24 degrees. Find the measure of the angles.

In the following exercises, translate to a system of equations and solve. The difference of two supplementary angles is 8 degrees. Find the measures of the angles.

In the following exercises, translate to a system of equations and solve. The difference of two supplementary angles is 88 degrees. Find the measures of the angles.