# Exercises: Solve Applications with Systems of Equations (4.4)

**Exercises: Translate to a System of Equations**

Instructions: For questions 1-14, translate to a system of equations and solve the system.

**1. The sum of two numbers is fifteen. One number is three less than the other. Find the numbers.**

**Solution**

The numbers are [latex]6[/latex] and [latex]9[/latex].

**2. The sum of two numbers is twenty-five. One number is five less than the other. Find the numbers.**

**3. The sum of two numbers is negative thirty. One number is five times the other. Find the numbers.**

**Solution**

The numbers are [latex]-5[/latex] and [latex]-25[/latex].

**4. The sum of two numbers is negative sixteen. One number is seven times the other. Find the numbers.**

**5. Twice a number plus three times a second number is twenty-two. Three times the first number plus four times the second is thirty-one. Find the numbers.**

**Solution**

The numbers are [latex]5[/latex] and [latex]4[/latex].

**6. 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.**

**7. 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.**

**Solution**

The numbers are [latex]2[/latex] and [latex]3[/latex].

**8. Twice a number plus three times a second number is negative one. The first number plus four times the second number is two. Find the numbers.**

**9. A married couple together earn [latex]$75\text{,}000[/latex]. The husband earns [latex]$15\text{,}000[/latex] more than five times what his wife earns. What does the wife earn?**

**Solution**

[latex]$10\text{,}000[/latex]

**10. During two years in college, a student earned [latex]$9\text{,}500[/latex]. The second year she earned [latex]$500[/latex] more than twice the amount she earned the first year. How much did she earn the first year?**

**11. Daniela invested a total of [latex]$50\text{,}000[/latex], some in a certificate of deposit (CD) and the remainder in bonds. The amount invested in bonds was [latex]$5000[/latex] more than twice the amount she put into the CD. How much did she invest in each account?**

**Solution**

She put [latex]$15\text{,}000[/latex] into a CD and [latex]$35\text{,}000[/latex] in bonds.

**12. Jorge invested [latex]$28\text{,}000[/latex] into two accounts. The amount he put in his money market account was [latex]$2\text{,}000[/latex] less than twice what he put into a CD. How much did he invest in each account?**

**13. In her last two years in college, Marlene received [latex]$42\text{,}000[/latex] in loans. The first year she received a loan that was [latex]$6\text{,}000[/latex] less than three times the amount of the second year’s loan. What was the amount of her loan for each year?**

**Solution**

The amount of the first year’s loan was [latex]$30\text{,}000[/latex] and the amount of the second year’s loan was [latex]$12\text{,}000[/latex].

**14. Jen and David owe [latex]$22\text{,}000[/latex] in loans for their two cars. The amount of the loan for Jen’s car is [latex]$2000[/latex] less than twice the amount of the loan for David’s car. How much is each car loan?**

**Exercises: Solve Direct Translation Applications**

Instructions: For questions 15-24, translate to a system of equations and solve.

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

**Solution**

Bethany is [latex]16[/latex] years old and Alyssa is [latex]28[/latex] years old.

**16. Robert is [latex]15[/latex] years older than his sister, Helen. The sum of their ages is sixty-three. Find their ages.**

**17. The age of Noelle’s dad is six less than three times Noelle’s age. The sum of their ages is seventy-four. Find their ages.**

**Solution**

Noelle is [latex]20[/latex] years old and her dad is [latex]54[/latex] years old.

**18. The age of Mark’s dad is [latex]4[/latex] less than twice Marks’s age. The sum of their ages is ninety-five. Find their ages.**

**19. 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?**

**Solution**

The small container holds [latex]20[/latex] gallons and the large container holds [latex]30[/latex] gallons.

**20. June needs [latex]48[/latex] 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?**

**21. Shelly spent [latex]10[/latex] minutes jogging and [latex]20[/latex] minutes cycling and burned [latex]300[/latex] calories. The next day, Shelly swapped times, doing [latex]20[/latex] minutes of jogging and [latex]10[/latex] 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?**

**Solution**

There were [latex]10[/latex] calories burned jogging and [latex]10[/latex] calories burned cycling.

**22. Drew burned [latex]1800[/latex] 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 [latex]3200[/latex] calories. How many calories did he burn per hour when playing basketball?**

**23. 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 [latex]$116[/latex]. Lisa bought two notebooks and three thumb dives for [latex]$68[/latex]. Find the cost of each notebook and each thumb drive.**

**Solution**

Notebooks are [latex]$4[/latex] and thumb drives are [latex]$20[/latex].

**24. Nancy bought seven pounds of oranges and three pounds of bananas for [latex]$17[/latex]. Her husband later bought three pounds of oranges and six pounds of bananas for [latex]$12[/latex]. What was the cost per pound of the oranges and the bananas?**

**Exercises: Solve Geometry Applications**

Instructions: For questions 25-40, translate to a system of equations and solve.

**25. The difference of two complementary angles is [latex]30[/latex] degrees. Find the measures of the angles.**

**Solution**

The measures are [latex]60[/latex] degrees and [latex]30[/latex] degrees.

**26. The difference of two complementary angles is [latex]68[/latex] degrees. Find the measures of the angles.**

**27. The difference of two supplementary angles is [latex]70[/latex] degrees. Find the measures of the angles.**

**Solution**

The measures are [latex]125[/latex] degrees and [latex]55[/latex] degrees.

**28. The difference of two supplementary angles is [latex]24[/latex] degrees. Find the measure of the angles.**

**29. The difference of two supplementary angles is [latex]8[/latex] degrees. Find the measures of the angles.**

**Solution**

[latex]94[/latex] degrees and [latex]86[/latex] degrees

**30. The difference of two supplementary angles is [latex]88[/latex] degrees. Find the measures of the angles.**

**31. The difference of two complementary angles is [latex]55[/latex] degrees. Find the measures of the angles.**

**Solution**

[latex]72.5[/latex] degrees and [latex]17.5[/latex] degrees

**32. The difference of two complementary angles is [latex]17[/latex] degrees. Find the measures of the angles.**

**33. Two angles are supplementary. The measure of the larger angle is four more than three times the measure of the smaller angle. Find the measures of both angles.**

**Solution**

The measures are [latex]44[/latex] degrees and [latex]136[/latex] degrees.

**34. Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles.**

**35. Two angles are complementary. The measure of the larger angle is twelve less than twice the measure of the smaller angle. Find the measures of both angles.**

**Solution**

The measures are [latex]34[/latex] degrees and [latex]56[/latex] degrees.

**36. Two angles are complementary. The measure of the larger angle is ten more than four times the measure of the smaller angle. Find the measures of both angles.**

**37. Wayne is hanging a string of lights [latex]45[/latex] feet long around the three sides of his rectangular patio, which is adjacent to his house. The length of his patio, the side along the house, is five feet longer than twice its width. Find the length and width of the patio.**

**Solution**

The width is [latex]10[/latex] feet and the length is [latex]25[/latex] feet.

**38. Darrin is hanging [latex]200[/latex] feet of Christmas garland on the three sides of fencing that enclose his rectangular front yard. The length, the side along the house, is five feet less than three times the width. Find the length and width of the fencing.**

**39. A frame around a rectangular family portrait has a perimeter of [latex]60[/latex] inches. The length is fifteen less than twice the width. Find the length and width of the frame.**

**Solution**

The width is [latex]15[/latex] feet and the length is [latex]15[/latex] feet.

**40. The perimeter of a rectangular toddler play area is [latex]100[/latex] feet. The length is ten more than three times the width. Find the length and width of the play area.**

**Exercises: Solve Uniform Motion Applications**

Instructions: For questions 41-52, translate to a system of equations and solve.

**41. Sarah left Minneapolis heading east on the interstate at a speed of [latex]60[/latex] mph. Her sister followed her on the same route, leaving two hours later and driving at a rate of [latex]70[/latex] mph. How long will it take for Sarah’s sister to catch up to Sarah?**

**Solution**

It took Sarah’s sister [latex]12[/latex] hours.

**42. College roommates John and David were driving home to the same town for the holidays. John drove [latex]55[/latex] mph, and David, who left an hour later, drove [latex]60[/latex] mph. How long will it take for David to catch up to John?**

**43. At the end of spring break, Lucy left the beach and drove back towards home, driving at a rate of [latex]40[/latex] mph. Lucy’s friend left the beach for home [latex]30[/latex] minutes (half an hour) later, and drove [latex]50[/latex] mph. How long did it take Lucy’s friend to catch up to Lucy?**

**Solution**

It took Lucy’s friend [latex]2[/latex] hours.

**44. Felecia left her home to visit her daughter driving [latex]45[/latex] mph. Her husband waited for the dog sitter to arrive and left home twenty minutes ([latex]\frac13[/latex] hour) later. He drove [latex]55[/latex] mph to catch up to Felecia. How long before he reaches her?**

**45. The Jones family took a [latex]12[/latex] mile canoe ride down the Indian River in two hours. After lunch, the return trip back up the river took three hours. Find the rate of the canoe in still water and the rate of the current.**

**Solution**

The canoe rate is [latex]5[/latex] mph and the current rate is [latex]1[/latex] mph.

**46. A motor boat travels [latex]60[/latex] miles down a river in three hours but takes five hours to return upstream. Find the rate of the boat in still water and the rate of the current.**

**47. A motor boat traveled [latex]18[/latex] miles down a river in two hours but going back upstream, it took [latex]4.5[/latex] hours due to the current. Find the rate of the motor boat in still water and the rate of the current. (Round to the nearest hundredth.).**

**Solution**

The boat rate is [latex]6.5[/latex] mph and the current rate is [latex]2.5[/latex] mph.

**48. A river cruise boat sailed [latex]80[/latex] miles down the Mississippi River for four hours. It took five hours to return. Find the rate of the cruise boat in still water and the rate of the current. (Round to the nearest hundredth.).**

**49. A small jet can fly [latex]1\text{,}072[/latex] miles in [latex]4[/latex] hours with a tailwind but only [latex]848[/latex] miles in [latex]4[/latex] hours into a headwind. Find the speed of the jet in still air and the speed of the wind.**

**Solution**

The jet rate is [latex]240[/latex] mph and the wind speed is [latex]28[/latex] mph.

**50. A small jet can fly [latex]1\text{,}435[/latex] miles in [latex]5[/latex] hours with a tailwind but only [latex]1\text{,}215[/latex] miles in [latex]5[/latex] hours into a headwind. Find the speed of the jet in still air and the speed of the wind.**

**51. A commercial jet can fly [latex]868[/latex] miles in [latex]2[/latex] hours with a tailwind but only [latex]792[/latex] miles in [latex]2[/latex] hours into a headwind. Find the speed of the jet in still air and the speed of the wind.**

**Solution**

The jet rate is [latex]415[/latex] mph and the wind speed is [latex]19[/latex] mph.

**52. A commercial jet can fly [latex]1\text{,}320[/latex] miles in [latex]3[/latex] hours with a tailwind but only [latex]1\text{,}170[/latex] miles in [latex]3[/latex] hours into a headwind. Find the speed of the jet in still air and the speed of the wind.**

**Exercises: Everyday Math**

**53. At a school concert, [latex]425[/latex]**** tickets were sold. Student tickets cost [latex]$5[/latex] each and adult tickets cost [latex]$8[/latex] each. The total receipts for the concert were [latex]$2\text{,}851[/latex].**

**Solve the system[latex]\left\{\begin{array}{c}s+a=425\\ 5s+8a=2\text{,}851\end{array}\right.[/latex] to find [latex]s[/latex], the number of student tickets and [latex]a[/latex], the number of adult tickets.
**

**Solution**

[latex]s=183,a=242[/latex]

**54. The first graders at one school went on a field trip to the zoo. The total number of children and adults who went on the field trip was [latex]115[/latex]. The number of adults was [latex]\frac{1}{4}[/latex] the number of children.**

**Solve the system [latex]\left\{\begin{array}{c}c+a=115\\ a=\frac{1}{4}c\end{array}\right.[/latex] to find [latex]c[/latex], the number of children and [latex]a[/latex], the number of adults.
**

**Exercises: Writing Exercises**

**55. Write an application problem similar to (Example 4.4.3) using the ages of two of your friends or family members. Then translate to a system of equations and solve it.**

**Solution**

Answers will vary.

**56. Write a uniform motion problem similar to (Example 4.4.8) that relates to where you live with your friends or family members. Then translate to a system of equations and solve it.**