1. Word sentence: The sum of three times a number and seven is one less than twice the same number.
Verb: “Is” indicates the location of the equal sign.

2. Word sentence: The sum of three times a number and seven is one less than twice the same number.
Phrase 1: “The sum of three times a number and seven” translates to 3n + 7.

3. Word sentence: The sum of three times a number and seven is one less than twice the same number.
Phrase 2: “One less than twice the same number” translates to 2n – 1.

4. Word sentence: The sum of three times a number and seven is one less than twice the same number.
Equation: 3n + 7 = 2n – 1.

5. Example 1
Find a number such that five times the sum of that number and – 3 is three less than the opposite of the number.
5(n – 3) = – n – 3
5n – 15 = – n – 3
5n + n – 15 = – n + n – 3
6n – 15 = – 3

6. Find a number such that five times the sum of that number and – 3 is three less than the opposite of the number.
6n – 15 = – 3
6n – = –
6n = 12 6
n = 2

7. Example
Twenty-five more than seven times a number is 46 more than four times the number.
7x + 25 = 4x + 46
x = 7

8. Example
The product of four and three less than a number is 15 less than five times the number.
4(n – 3) = 5n – 15
n = 3

9. Steps in Solving a Word Problem
Represent an unknown quantity with a variable.
When necessary, represent other conditions in the problem in terms of the variable.

10. Steps in Solving a Word Problem
3. Identify two equal quantities in the problem.
4. Write and solve an equation.
5. Check the answer.

11. Example 2
The sum of two consecutive integers is 173. Find the integers.
Let x = the smaller integer.
Let x + 1 = the larger integer.
x + (x + 1) = 173
(x + x) + 1 = 173
2x + 1 = 173

12. The sum of two consecutive integers is 173. Find the integers.
2x + 1 = 173
2x + 1 – 1 = 173 – 1
2x = 172 2
x = 86
x + 1 = 87

13. 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 2

14. Example 3
Find three consecutive odd integers such that three times the third is two more than five times the first.
Let x = the smallest integer.
Let x + 2 = the next consecutive odd integer.
Let x + 4 = the largest integer.

15. Find three consecutive odd integers such that three times the third is two more than five times the first.
3(x + 4) = 5x + 2
3x + 12 = 5x + 2
3x – 3x + 12 = 5x – 3x + 2
12 = 2x + 2
12 – 2 = 2x + 2 – 2
10 = 2x

16. Find three consecutive odd integers such that three times the third is two more than five times the first.
10 = 2x 2
x = 5
x + 2 = 7
x + 4 = 9

17. Example 4
If a ship is traveling at 20 knots (nautical miles per hour), how long will it take to travel 55 nautical miles?
r t = d
20t = 55 20
t = 2.75 hr.

18. Example 5
What is the speed of the wind if a plane capable of flying 400 mi./hr. takes 3 hr. to travel 1,050 mi. into a headwind?
(r – w) t = d
(400 – w) 3 = 1,050
1200 – 3w = 1,050
1200 – 1200 – 3w = 1,050 – 1200

19. What is the speed of the wind if a plane capable of flying 400 mi. /hr
What is the speed of the wind if a plane capable of flying 400 mi./hr. takes 3 hr. to travel 1,050 mi. into a headwind?
1200 – 1200 – 3w = 1,050 – 1200
– 3w = – 150
– 3
w = 50 mi./hr.

20. Example 6
Luke and Andrew paddled their canoe 22 mi. down the river in 4 hr. If the river’s current averages 1.5 mi./hr., what is their average rate in still water?
(r + c) t = d
(r + 1.5) 4 = 22
4r + 6 = 22

21. Luke and Andrew paddled their canoe 22 mi. down the river in 4 hr
Luke and Andrew paddled their canoe 22 mi. down the river in 4 hr. If the river’s current averages 1.5 mi./hr., what is their average rate in still water?
4r + 6 – 6 = 22 – 6
4r = 16 4
r = 4 mi./hr.

22. Exercise
Find two consecutive even integers such that twice the larger is four more than twice the smaller.

23. Exercise
Find two consecutive integers that five times the smaller equals three times the larger.

24. Exercise
After rowing down the stream in 4 hr., the students were able to row back up the stream in 6 hr. If the stream flows at 1 mi./hr., how fast can the students row their boat in still water?

25. Exercise
Solve the formula d = r t for the variable t and then for the variable r.