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Topic 2: Algebraic Expressions/Equations Lesson 6
Corresponds to Section 1.5
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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
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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
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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¦β2, β1, 0, 1, 2, β¦ 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¦β2, β1, 0, 1, 2, β¦} 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}
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Example 1: Replacement Set
Find the solution set of the equation 2π+5=13 if the replacement set is {2, 3, 4, 5, 6} 1. Create a Table 2. Replace π with each value in the replacement set Solution Set: {4} π ππ+π=ππ 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
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Example 2: Solving Equations with One Variable
Solve the equation 7β 4 2 β10 +π=10 1. 7β 16β10 +π=10 2. 7β 6 +π=10 3. 1+π=10 4. π=9 5. This equation has one unique solution of 9.
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Example 3: Solving Equations with One Variable
Solve the equation π =5π+(10β3) 1. π 5 +6=5π+(7) 2. 5π+6=5π+7 3. No matter what value we substitute for π, the equation will never be true. Thus, the equation has no solution.
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Example 4: Solving Equations with One Variable
Solve the equation 2β5β8 3β+6 = 2β+β +6 2 1. 10β8 3β+6 = 2β+β +6 2 2. 2 3β+6 = 2β+β +6 2 3. 6β+12= 2β+β +6 2 4. 6β+12= 3β +6 2 5. 6β+12=6β+12 6. Since both sides are the same, this equationβs solution is all real numbers. In other words, any number will make it true.
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Identity An equation that is true for every value of the variable.
EX: 6β+12=6β+12
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Example 5: Solve the equation 3 π+1 β5=3πβ2 1. 3π+3β5=3πβ2
2. 3πβ2=3πβ2 3. Solution Set: All Real Numbers 4. This is an identity
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Example 6: Solve the equation 2π+ 2 3 β5 =10 5β2 +π 12Γ·6
1. 2π+ 8β5 =10 5β2 +π(12Γ·6) 2. 2π+ 3 =10 5β2 +π(12Γ·6) 3. 2π+3=10 3 +π(12Γ·6) 4. 2π+3=10 3 +π(2) 5. 2π+3=30+2π 6. 2π+3=2π+30 7. Solution Set: No Solutions
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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 πΆ be the total cost and π be the number of movies. C = 1.5m + 10
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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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