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Unit 1: Relationships between Quantities and Reasoning with Equations Lesson 5- Using Properties to Solve Equations Objectives: I can identify the properties of real numbers and properties of equality. I can use the properties to justify each step of a solution and solve equations. Warm Up: State the inverse. Addition - Subtraction- Squaring- Multiplication- Division- Square root- Properties of Real Numbers: Whenever you simplify or evaluate an expression, you use properties of real numbers. You probably use these properties without even realizing it. They are what justify many of the steps you take when working with expressions and equations. Here are some of the important properties of real numbers… Property Associative Addition Multiplication a+(b + c) = (a + b) + c a (bc) = (ab)c Commutative a+b = b+a ab = ba Identity a+0=a a1 = a Inverse a + (-a) = 0 a (1/a) = 1 Distributive a (b + c) = ab + ac *Note the multiplicative inverse of a number is also called the reciprocal and the additive inverse of a number is also called the opposite. You try! Identify the property. 3(5x – 1) 5+0=5 3(42) = (34)2 61 = 6 3+2=2+3 8 + (-8) = 0 Properties are also useful for manipulating equations in order to find solutions. Look at the next page for the properties of equality. The properties of equality listed below can be used to isolate variables and find their values. Properties of Equality Addition Property If a = b, then a+c = b+c. Subtraction Property If a = b, then a - c = b - c. Multiplication Property If a = b, then ac = bc. 𝑎 𝑏 Division Property If a = b and c ≠ 0, then = . 𝑐 𝑐 Reflexive Property a=a Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Substitution Property If a = b, then b can replace a in any expression. Felicia wrote the steps shown below while solving the equation below. Use the properties of real numbers and properties of equality to justify each step in Felicia’s solution. 3(4 + x) = -3. Step 1: 12 + 3x = -3 ____________________ Step 2: 3x + 12 = -3 ____________________ Step 3: 3x = -15 ____________________ Step 4: x = -5 ____________________ Think… Can you think of a way to check to see if the solution is correct? How? Brandon was given the equation 36 = 5x + (3y – 7x) and was asked to write it in slope-intercept form, y = mx + b. The steps Brandon took are shown below. Use properties to justify each step in the solution. 36= 5x + (3y – 7x) Step 1: 36 = 5x + (-7x + 3y) ____________________ Step 2: 36 = [5x + (-7x)] + 3y ____________________ Step 3: 36 = -2x + 3y ____________________ Step 4: 2x + 36 = 3y ____________________ Step 5: 2 3 𝑥 + 12 = 𝑦 Step 6: 𝑦 = 2 3 𝑥 + 12 ____________________ ____________________ Kevin was given the equation 3x + 5y = 12 and y = 2 – x. He used the following steps to find the x-value of the coordinate pair that satisfies both equation. 3x + 5y = 12 Step 1: 3x + 5(2 – x) = 12 _____________________ Step 2: 3x + 10 – 5x = 12 _____________________ Step 3: -2x + 10 = 12 ______________________ Step 4: -2x = 2 ______________________ Step 5: x = -1 ______________________ To find the y-value, Kevin took the following steps. y=2–x Step 1: y = 2 – (-1) Step 2: y = 3 Which property justifies Step 1? ________________________ Solving Equations: Now that we have looked at how the properties are used to justify the steps of a solution, let’s solve! We can follow the following 5 step process when we are solving equations… Step 1: Distribute Step 2: Add like terms on each side of the equation Step 3: Get the variables on one side of the equation Step 4: Get rid of the constant by using the additive inverse Step 5: Get rid of the coefficient by using the multiplicative inverse Let’s try a few… 2x – 5 = 7 3(2x + 4) = -12 -x + 4x + 2 = 8 6x + 15 = 3x + 8 3(2x + 5) = 4x + 7 – x + 1 5𝑦 − 2 =3 4 You try! 3x – 5 = 16 3x + 5 – 2x = 6x – 10 6x + 2(2x + 3) = 16 3 – 2(x + 4) = -3(4x – 5) ==================================================== Write your own equation that requires multiple steps to solve. Solve it and justify each step of the solution. Name: _________________________ Unit 1 Lesson 5: Using Properties to Solve Equations Identify the property of real numbers that is demonstrated. 1) 12 + 0 = 12 2) 9 + d = d + 9 3) 7(2 – p) = 14 – 7p 4) 6p = 16p Identify the property of equality that is demonstrated. 5) 13x7 = 13x7 6) If z = 12 and 12 = 34, then z = 34. 7) If 12t = 5s, then 5s = 12t. 8) If n = 0.25p and 3p + 2n = 14, then 3.5p = 14. Use properties to justify each step taken to solve the equations. 9) 8 + 7x – 8 = 49 8 – 8 + 7x = 49 ______________________ 0 + 7x = 49 ______________________ 7x = 49 ______________________ x=7 ______________________ Solve for x. Use properties to justify the steps you use. 10) x + 6 = 8 11) y = -5 x=y Property: ________________ Property: _________________ 12) The associative property of multiplication states that (a x b) x c = a x (b x c). How would you express this property in words? 13) Alexis solved the equation 5x = 4 in the two ways shown below. Method 1 5x = 4 5x ÷ 5 = 4÷5 x= 4 5 Method 2 5x = 4 1 5 ∙ 5𝑥 = 4 ∙ x= 1 5 4 5 Use properties to justify the steps in each of Alexis’s solution methods. What do Alexis’s methods tell you about the relationship between the properties she used? 14) Solve. 2(-x + 4) = 5x – 4 15) Solve. 𝑥+3 =5 2