An Introduction to Equations Section 1-8

Goals Goal Rubric To solve equations using tables and mental math. Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems.

Vocabulary Equation Open sentence Solution of an equation

Definition Equation – A mathematical sentence that states one expression is equal to a second expression. mathematical sentence that uses an equal sign (=). (value of left side) = (value of right side) An equation is true if the expressions on either side of the equal sign are equal. An equation is false if the expressions on either side of the equal sign are not equal. Examples: 4x + 3 = 10 is an equation, while 4x + 3 is an expression. 5 + 4 = 9 True Statement 5 + 3 = 9 False Statement

Equation or Expression In Mathematics there is a difference between a phrase and a sentence. Phrases translate into expressions; sentences translate into equations or inequalities. Phrases Expressions Sentences Equations or Inequalities

Definition Open Sentence – an equation that contains one or more variables. An open sentence is neither true nor false until the variable is filled in with a value. Examples: Open sentence: 3x + 4 = 19. Not an open sentence: 3(5) + 4 = 19.

Example: Classifying Equations Is the equation true, false, or open? Explain. 3y + 6 = 5y – 8 Open, because there is a variable. 16 – 7 = 4 + 5 True, because both sides equal 9. 32 ÷ 8 = 2 ∙ 3 False, because both sides are not equal, 4 ≠ 6.

Your Turn: Is the equation true, false, or open? Explain. 17 + 9 = 19 + 6 False, because both sides are not equal, 26 ≠ 25. 4 ∙ 11 = 44 True, because both sides equal 44. 3x – 1 = 17 Open, because there is a variable.

Definition Solution of an Equation – is a value of the variable that makes the equation true. A solution set is the set of all solutions. Finding the solutions of an equation is called solving the equation. Examples: x = 5 is a solution of the equation 3x + 4 = 19, because 3(5) + 4 = 19 is a true statement.

Example: Identifying Solutions of an Equation Is m = 2 a solution of the equation 6m – 16 = -4? 6m – 16 = -4 6(2) – 16 = -4 12 – 16 = -4 -4 = -4 True statement, m = 2 is a solution.

Your Turn: Is x = 5 a solution of the equation 15 = 4x – 4? No, 15 ≠ 16. False statement, x = 5 is not a solution.

Procedure for Writing an Equation A PROBLEM SOLVING PLAN USING MODELS VERBAL MODEL Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know. Ask yourself what you need to know to solve the problem. Then write a verbal model that will give you what you need to know. LABELS Assign labels to each part of your verbal problem. Assign labels to each part of your verbal problem. ALGEBRAIC MODEL Use the labels to write an algebraic model based on your verbal model. Use the labels to write an algebraic model based on your verbal model.

Writing an Equation You and three friends are having a dim sum lunch at a Chinese restaurant that charges $2 per plate. You order lots of plates. The waiter gives you a bill for $25.20, which includes tax of $1.20. Write an equation for how many plates your group ordered. SOLUTION Understand the problem situation before you begin. For example, notice that tax is added after the total cost of the dim sum plates is figured.

– – • = = = The equation is 2p = 24. Cost per plate = 2 p Writing an Equation VERBAL MODEL Cost per plate Number of plates • = – Bill Tax LABELS Cost per plate = 2 (dollars) ALGEBRAIC MODEL p Number of plates = (plates) Amount of bill = 25.20 (dollars) Tax = 1.20 (dollars) 2 = p 25.20 1.20 – 2p = 24.00 The equation is 2p = 24.

Your Turn: JET PILOT A jet pilot is flying from Los Angeles, CA to Chicago, IL at a speed of 500 miles per hour. When the plane is 600 miles from Chicago, an air traffic controller tells the pilot that it will be 2 hours before the plane can get clearance to land. The pilot knows the speed of the jet must be greater then 322 miles per hour or the jet could stall. Write an equation to find at what speed would the jet have to fly to arrive in Chicago in 2 hours?

Solution At what speed would the jet have to fly to arrive in Chicago in 2 hours? SOLUTION You can use the formula (rate)(time) = (distance) to write a verbal model. VERBAL MODEL Speed of jet Distance to travel • Time = LABELS Speed of jet = x (miles per hour) ALGEBRAIC MODEL Time = 2 (hours) Distance to travel = 600 (miles) 2 x = 600 2x = 600

Example: Use Mental Math to Find Solutions What is the solution to the equation? Use mental math. 12 – y = 3 Think: What number subtracted from 12 equals 3. Solution: 9. Check: 12 – (9) = 3, 3 = 3 is a true statement, therefore 9 is a solution.

Your Turn: What is the solution to the equation? Use mental math. x + 7 = 13 6 x/6 = 12 72

Joke Time What do you call a guy with a rubber toe? ROBERTO. What do you get if you cross a dinosaur with a pig? Jurassic Pork. What do you get when you put a bomb and a dinosaur together? Dino-mite.

Assignment 1.8 Exercises Pg. 64 – 66: #8 – 32 even, 48 – 52 even, 56 – 74 even