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Topic 2: Algebraic Expressions/Equations Lesson 6

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Presentation on theme: "Topic 2: Algebraic Expressions/Equations Lesson 6"β€” Presentation transcript:

1 Topic 2: Algebraic Expressions/Equations Lesson 6
Corresponds to Section 1.5

2 Learning Targets Define the following vocabulary terms: open sentence, equation, solving an equation, solution, replacement set, set, element, solution set, identity Use a replacement set to solve an equation Solve an equation with one variable Solve an equation with multiple variables Create an equation with a unique solution, no solution, and many solutions

3 Open Sentence vs Equation
Open Sentence: Mathematical statement that contains expressions and symbols EX: 3π‘₯+7 Equation: Mathematical sentence that contains an equal sign EX: 3π‘₯+7=13 Key Question: Compare and Contrast Expressions and Equations. Connect it to a non-mathematical idea

4 Components of an Equation
Solving an Equation: The process of finding a value(s) for a variable that makes the sentence true. EX: π‘₯+5=8 Replacement Set: A set of numbers where replacements of a variable may be chosen EX: {2, 4, 6, 8} Set: a collection of objects or numbers EX: {2, 5, 9, 10, 11} EX: β€¦βˆ’2, βˆ’1, 0, 1, 2, … Element: each object/member in the set EX: 2 is an element of the set {2, 5, 9, 10, 11} EX: -10 is an element of the set {β€¦βˆ’2, βˆ’1, 0, 1, 2, …} Equation Solution: A specific value that replaces the variable to make the sentence true. EX: π‘₯=3 Solution Set: Set of elements from the replacement set that make an equation true. EX: {3}

5 Example 1: Replacement Set
Find the solution set of the equation 2π‘ž+5=13 if the replacement set is {2, 3, 4, 5, 6} 1. Create a Table 2. Replace π‘ž with each value in the replacement set Solution Set: {4} 𝒒 πŸπ’’+πŸ“=πŸπŸ‘ True or False? 2 2 2 +5=9β‰ 13 False 3 2 3 +5=11β‰ 13 4 2 4 +5=13 True 5 2 5 +5=15β‰ 13 6 2 6 +5=17β‰ 13

6 Example 2: Solving Equations with One Variable
Solve the equation 7βˆ’ 4 2 βˆ’10 +𝑛=10 1. 7βˆ’ 16βˆ’10 +𝑛=10 2. 7βˆ’ 6 +𝑛=10 3. 1+𝑛=10 4. 𝑛=9 5. This equation has one unique solution of 9.

7 Example 3: Solving Equations with One Variable
Solve the equation 𝑛 =5𝑛+(10βˆ’3) 1. 𝑛 5 +6=5𝑛+(7) 2. 5𝑛+6=5𝑛+7 3. No matter what value we substitute for 𝑛, the equation will never be true. Thus, the equation has no solution.

8 Example 4: Solving Equations with One Variable
Solve the equation 2βˆ™5βˆ’8 3β„Ž+6 = 2β„Ž+β„Ž +6 2 1. 10βˆ’8 3β„Ž+6 = 2β„Ž+β„Ž +6 2 2. 2 3β„Ž+6 = 2β„Ž+β„Ž +6 2 3. 6β„Ž+12= 2β„Ž+β„Ž +6 2 4. 6β„Ž+12= 3β„Ž +6 2 5. 6β„Ž+12=6β„Ž+12 6. Since both sides are the same, this equation’s solution is all real numbers. In other words, any number will make it true.

9 Identity An equation that is true for every value of the variable.
EX: 6β„Ž+12=6β„Ž+12

10 Example 5: Solve the equation 3 𝑏+1 βˆ’5=3π‘βˆ’2 1. 3𝑏+3βˆ’5=3π‘βˆ’2
2. 3π‘βˆ’2=3π‘βˆ’2 3. Solution Set: All Real Numbers 4. This is an identity

11 Example 6: Solve the equation 2𝑑+ 2 3 βˆ’5 =10 5βˆ’2 +𝑑 12Γ·6
1. 2𝑑+ 8βˆ’5 =10 5βˆ’2 +𝑑(12Γ·6) 2. 2𝑑+ 3 =10 5βˆ’2 +𝑑(12Γ·6) 3. 2𝑑+3=10 3 +𝑑(12Γ·6) 4. 2𝑑+3=10 3 +𝑑(2) 5. 2𝑑+3=30+2𝑑 6. 2𝑑+3=2𝑑+30 7. Solution Set: No Solutions

12 Example 7: 2 Variables(pg 35 #5)
Mr. Hernandez pays $10 a month for movies delivered by mail. He can also rent movies in the store for $1.50 per title. A) Write an equation that describes the totally amount Mr. H pays each month. Let 𝐢 be the total cost and π‘š be the number of movies. C = 1.5m + 10

13 Example 7 continued(pg 35 #5)
Mr. Hernandez pays $10 a month for movies delivered by mail. He can also rent movies in the store for $1.50 per title. B) How much did Mr. H spend this month if he rents 3 movies from the store? C = 1.5(3) + 10 = $14.50

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