1. Lesson 14: from problems to equations

2. Many problems can be transformed into equations
Problems to equations
Many problems can be transformed into equations
To transform them you must:

3. Step 1 – Identify the unknown quantity or quantities
Problems to equations
Step 1 – Identify the unknown quantity or quantities

4. Problems to equations
The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine Claude and Jean’s ages.
What THINGS are we talking about?????

5. Problems to equations
Step 2 – Represent each unknown with a variable or algebraic expression

6. 𝑥 2𝑥+10 Problems to equations Jean Claude
The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine Claude and Jean’s ages.
Claude
Jean 𝑥
2𝑥+10

7. Problems to equations
Step 3 – Form an equation

8. 𝑥 + 2𝑥+10 =52 𝑥 2𝑥+10 Problems to equations Jean Claude
The sum of Claude and Jean’s ages is 52 years. Jean is 10 years older than twice Claude’s age. Determine Claude and Jean’s ages. 𝑥 +
2𝑥+10
Claude
Jean
=52 𝑥
2𝑥+10

9. 𝑥 + 2𝑥+10 =52 3𝑥 + 10 =52 − 10 − 10 𝑥 =14 3𝑥 =42 Step 4 – Solve 3 3
Problems to equations
Step 4 – Solve 𝑥 +
2𝑥+10
=52 3𝑥
+ 10
=52
− 10
− 10 𝑥
=14 3𝑥
=42 3 3

10. Problems to equations
Step 5 – Express and verify the solution 14 38
Claude
Jean 𝑥
=14 𝑥
2𝑥+10 14
28+10

11. Problems to equations
Verify
14+38 52

12. 𝑥 𝑥−7 Problems to equations Petra Ivan
2. Petra is 7 years younger than her brother Ivan. Together, their ages total 43 years. How old is Petra?
Petra
Ivan 𝑥
𝑥−7

13. 2𝑥 =50 𝑥 + 𝑥−7 =43 2𝑥 −7 =43 𝑥 =25 2 2 +7 +7 Problems to equations
2. Petra is 7 years younger than her brother Ivan. Together, their ages total 43 years. How old is Petra? 2𝑥
=50 𝑥 +
𝑥−7
=43 2𝑥 2 −7
=43 2 +7 +7 𝑥
=25

14. 𝑥 =25 𝑥 𝑥−7 25−7 18 Problems to equations Petra Ivan
2. Petra is 7 years younger than her brother Ivan. Together, their ages total 43 years. How old is Petra?
Petra
Ivan 𝑥
=25 𝑥
𝑥−7
25−7 18

15. 𝑥 𝑥+9 Problems to equations Yellow Blue
3. There are 43 marbles in a box. Some of the marbles are yellow, and the rest are blue. There are 9 more yellow marbles than blue marbles. How many blue marbles are in the box?
Yellow
Blue 𝑥
𝑥+9

16. Problems to equations
Yellow
Blue 𝑥
=43
𝑥+9 + 𝑥
𝑥+9
=43 2𝑥
+ 9 17 −9 −9 2𝑥
=34 𝑥
=17 2 2
There are 17 blue marbles in the box

17. 2∙𝑥 +3 = −4 3∙𝑥 2𝑥+3 = 3𝑥−4 Problems to equations
4. Twice a number increased by three is the same as four less than three times the same number. What is the number? −4
3∙𝑥
2𝑥+3 =
3𝑥−4

18. 2𝑥+3 = 3𝑥−4 2𝑥 = 3𝑥 −7 𝑥 = 7 −𝑥 = −7 − 3 − 3 −3𝑥 −3𝑥 −1 −1
Problems to equations
2𝑥+3 =
3𝑥−4
− 3
− 3 2𝑥 = 3𝑥 −7
−3𝑥
−3𝑥 𝑥 = 7 −𝑥 = −7 −1 −1

19. 𝐴=60 𝑐𝑚 2 𝐴= 𝑏∙ℎ 2 𝑏=15𝑐𝑚 ℎ= ? Problems to equations
5. A triangle has an area of 60 𝑐𝑚 2 . If one of its bases is 15cm long, determine the corresponding height.
𝐴=60 𝑐𝑚 2
𝐴= 𝑏∙ℎ 2 h
𝑏=15𝑐𝑚
ℎ= ? b

20. 𝐴=60 𝑐𝑚 2 𝐴= 𝑏∙ℎ 2 ∙2 60= 15∙ℎ 2 ∙2 𝑏=15𝑐𝑚 ℎ= ? 120 = 15∙ℎ 15 15 8 = ℎ
Problems to equations
𝐴=60 𝑐𝑚 2
𝐴= 𝑏∙ℎ 2 ∙2
60= 15∙ℎ 2 ∙2
𝑏=15𝑐𝑚
ℎ= ?
120 =
15∙ℎ 15 15 h 8 = ℎ b

21. =147 𝑥 𝑥+2 𝑥+4 + + Problems to equations n1 n2 n3
6. The sum of 3 consecutive odd numbers is What are these numbers? n1 n2 n3
=147 𝑥
𝑥+2
𝑥+4 + +

22. Problems to equations 𝑥 +
𝑥+2
𝑥+4
=147 + 3𝑥
+ 6
=147 −6 −6 𝑥
=47 3𝑥
=141 3 3

23. 𝑥 𝑥+2 𝑥+4 47 47+2 47+4 49 51 Problems to equations n1 n2 n3
6. The sum of 3 consecutive odd numbers is What are these numbers? n1 n2 n3
The numbers are 47, 49 and 51 𝑥
𝑥+2
𝑥+4 47
47+2
47+4 49 51

24. 𝑥 2𝑥+3 = +3 Problems to equations n1 n2 2∙𝑛𝑢𝑚𝑏𝑒𝑟 2 𝑛𝑢𝑚𝑏𝑒𝑟 1
7. A number is 3 more than twice another number. The difference of these numbers is 75. What are the two numbers? n1 n2 𝑥
2𝑥+3

25. (2𝑥+3) − (𝑥) =75 2𝑥+3 − 𝑥 =75 𝑥 𝑥 +3 =75 =72 − −3 −3
Problems to equations
7. A number is 3 more than twice another number. The difference of these numbers is 75. What are the two numbers? −
=75
(2𝑥+3) −
(𝑥)
=75
2𝑥+3
− 𝑥
=75 𝑥 𝑥 +3
=75
=72 −3 −3

26. 147 𝑥 2𝑥+3 2(72)+3 72 144+3 Problems to equations n1 n2
7. A number is 3 more than twice another number. The difference of these numbers is 75. What are the two numbers? n1 n2
147
2𝑥+3 𝑥
2(72)+3 72
The numbers are 147 and 72
144+3