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Lesson 4.2 Rational Numbers & Equations Solving One-Step Equations Warm-Up Evaluate each expression for the given value(s). 1. 3x – 2 when x = 5 13 2. 7 – 6y when y = –2 19 p 3. – 12 when p = 10 2 –7 1 4. d + (–2) when d = 12 4 5. 9m + 3n when m = –1 and n = 2 1 –3 Lesson 4.2 Solving One-Step Equations Use inverse operations to solve one-step equations. The Properties of Equality For any numbers a, b and c: Subtraction Property of Equality If a = b, then a – c = b – c Addition Property of Equality If a = b, then a + c = b + c Multiplication Property of Equality If a = b, then ac = bc Division Property of Equality a b If a = b, then c c Explore! Introduction to Equation Mats Step 1 If you do not have an equation mat, draw one like the one seen below on a blank sheet of paper. Step 2 On your equation mat, place a variable cube on one side with 3 negative integer chips. On the other side of the mat, place 5 positive integer chips. This represents the equation x − 3 = 5. Explore! Introduction to Equation Mats Step 3 In order to get the variable by itself, you must cancel out the 3 negative integer chips with the variable. Use zero pairs to remove the chips by adding three positive integer chips to the left side of the mat. Whatever you add on one side of the mat, add on the other side of the mat. This is using the Addition Property of Equality. How many chips are on the right side of the mat? What does this represent? Step 4 Clear your mat and place chips and variable cubes on the mat to represent the equation 4x = −8. Draw this on your own paper. Step 5 Divide the integer chips equally among the variable cubes. This is using the Division Property of Equality. Each variable cube is equal to how many integer chips? Write your answer in the form x = ___. Good to Know! You can solve equations using inverse operations to keep your equation balanced. Inverse operations are operations that undo each other, such as addition and subtraction. Even though you may be able to solve many one-step equations mentally, it is important that you show your work. The equations you will be solving in later lessons and in future math classes will become more complex. Drawing a vertical line through the equals sign can help you stay organized. Whatever is done on one side of the line to cancel out a value must be done on the other side. Example 1 Solve each equation. Show your work and check your solution. a. x + 13 = 41 The vertical line can help you stay organized. The inverse operation of addition is subtraction. Subtract 13 from both sides of the equation to isolate the variable. Check the answer by substituting the solution into the original equation for the variable. x + 13 = 41 –13 –13 x = 28 (28) + 13 = 41 41 = 41 Example 1 Continued… Solve each equation. Show your work and check your solution. b. 6m = 27 Divide both sides of the equation by 6. Check the solution. 6m = 27 6 6 1 m = 4.5 or m = 4 2 6(4.5) = 27 27 = 27 Example 1 Continued… Solve each equation. Show your work and check your solution. y c. 4 3 The variable can be on either side of the equation. Multiply both sides of the equation by 3. y 3 4 3 3 12 y Check the solution. (12) 4 3 4 4 ? Example 2 The meteorologist on Channel 3 announced that a record high temperature has been set in Kirkland today. The new record is 2.8º F more than the old record. Today’s high temperature was 98.3º F. What was the old record? Let x represent the old record temperature. The equation witch represents the situation is: Subtract 2.8 from each side of the equation to isolate the variable. Check The old record high temperature in Kirkland was 95.5º F. x + 2.8 = 98.3 x + 2.8 = 98.3 – 2.8 – 2.8 x = 95.5 95.5 + 2.8 98.3 98.3 = 98.3 Communication Prompt How are inverse operations useful when solving equations? Formative Assessment 4.2 Solve each equation. Show your work and check your solution. 1. x + 39 = 150 x = 111 2. d 3 5 d = –15 3. 12.9 = y – 14.2 y = 27.1 4. –5t = 55 t = –11