Lesson 14: from problems to equations

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

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

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?????

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

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

Problems to equations Step 3 – Form an equation

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

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

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

Problems to equations Verify 14+38 52

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

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

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

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

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

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

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

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

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

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

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

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 147. 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

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

(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

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