Slide 1 / 138 Slide 2 / 138 6th Grade Equations & Inequalities 2015-12-01 www.njctl.org Slide 3 / 138 Slide 4 / 138 Table of Contents Click on a topic to Equations and Identities · go to that section. Tables · Determining Solutions of Equations · Solving an Equation for a Variable · Equations and Identities Solving One Step Addition & Subtraction Equations · Solving One Step Multiplication & Division Equations · Writing Equations · Writing Simple Inequalities · Solutions to Simple Inequalities · Graphing Solution Sets to Simple Inequalities Return to Table · of Contents Glossary & Standards · Slide 5 / 138 Slide 6 / 138 Equations and Identities Equations and Identities What is an equation? An equation is created when two expressions are set equal to one another such that they are equal for some values of How is it different than an identity? their variables, but not for all. If they are equal for all values of their variables, then that is Discuss in your groups. an identity, not an equation. So, not all mathematical statements which include an equal sign are equations...some are identities.
Slide 7 / 138 Slide 8 / 138 Equations and Identities Equations and Identities Here are some equations: Here are some identities : x = 5t 2 + 3 = 5 v = 7 + x 9 - 2 = 7 a = x - 8 2x = 2x In all these cases, the variables are interdependent. They are only true for certain sets of variables. 9/3 = 3 Changing the value of the variable on the right side of the These are always true...there are no values that can be equation, changes the possible values of the variables on the assigned to the variables for which these would be untrue. left side...and vice versa. These are equations (not identities) since knowing the value of one variable changes the possible value(s) of the other(s). Slide 9 / 138 Slide 10 / 138 Writing Equations Write an equation in words. Then translate that into a mathematical equation. Tim is eight years older than Kathy. Write an equation for Kathy's age. Tables Tim's age is equal to Kathy's plus 8 click for equation in words T = K + 8 click for mathematical equation Return to Table of Contents Slide 11 / 138 Slide 12 / 138 Writing Equations Writing Equations Write an equation in words. Write an equation in words. Then translate that into a mathematical equation. Then translate that into a mathematical equation. Tim is eight years older than Kathy. Write an equation for Bob is 6 inches less than twice the height of Fred. Write an Kathy's age. equation for Bob's height. click for equation in words Bob's height equals double Fred's height less 6 Tim's age is equal to Kathy's plus 8 click for equation in words T = K + 8 click for mathematical equation click for mathematical equation B = 2F - 6
Slide 13 / 138 Slide 14 / 138 Writing Equations Tables and Expressions Write an equation in words. Let's use this table to find some solutions to the equation s = d/t; where s represents speed (in meters/second), d represents Then translate that into a mathematical equation. distance (in meters) and t represents time (in seconds). Math Practice Speed is equal to the distance traveled in an amount of time. s = d/t We've entered the distance d (m) t (s) s (m/s) traveled and the time it took 30 2 to travel that distance in two click for equation in words Speed equals distance divided by time of the columns. 60 4 90 6 Use the equation (s = d/t) to 120 2 find the speeds and fill in the click for mathematical equation s = d/t blank column. 240 4 360 6 Slide 15 / 138 Slide 16 / 138 Tables and Expressions Note that in the first three sets of answers, the object was moving at a speed of 15 m/s. The final three sets of answers are for an object traveling four Determining Solutions of times faster, at 60 m/s. s = d/t Equations But, in all cases, knowing d (m) t (s) s (m/s) the value of two of the three 30 2 15 variables determines the 60 4 15 values of the third. 90 6 15 120 2 60 240 4 60 Return to Table 360 6 60 of Contents Slide 17 / 138 Slide 18 / 138 Determining the Solutions of Equations Determining the Solutions of Equations Which of the following is a solution of the equation from the A solution to an equation is a number that makes the equation solution set ? true. x + 7 = 9 {2, 3, 4, 5} In order to determine if a number is a solution, replace the variable with the number and evaluate the equation. Write the equation four times. Each time replace x with one of the possible solutions and simplify to see if it is true. If the number makes the equation true, it is a solution. If the number makes the equation false, it is not a solution. 2 + 7 = 9 3 + 7 = 9 4 + 7 = 9 5 + 7 = 9 9 = 9 10 = 9 11 = 9 12 = 9 Yes No No No Answer: 2 is the solution to x + 7 = 9
Slide 19 / 138 Slide 20 / 138 Determining the Solutions of Equations 1 Which of the following is a solution to the equation? Which of the following is a solution of the equation? x + 17 = 21 {2, 3, 4, 5} y - 12 = 8 {17, 18, 19, 20} Write the equation four times. Each time replace y with one of the possible solutions and simplify to see if it is true. 17 - 12 = 8 18 - 12 = 8 19 - 12 = 8 20 - 12 = 8 5 = 8 6 = 8 7 = 8 8 = 8 No No No Yes Answer: 20 is the solution to y - 12 = 8 Slide 21 / 138 Slide 22 / 138 2 Which of the following is a solution to the equation? 3 Which of the following is a solution to the equation? m - 13 = 28 {39, 40, 41, 42} 12b = 132 {9, 10, 11, 12} Slide 23 / 138 Slide 24 / 138 4 Which of the following is a solution to the equation? 3.5 + d = 7.5 {2.5, 3, 3.5, 4} Solving an Equation for a Variable Return to Table of Contents
Slide 25 / 138 Slide 26 / 138 The Rules for Solving Equations The Rules Like in any game there are a few rules. Here are the three rules. Let's examine them, one at a time. There are three rules which will allow you to solve any one-step 1. To "undo" a mathematical operation, you must perform the equation. inverse operation. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other. 3. You can always switch the left and right sides of an equation. Slide 27 / 138 Slide 28 / 138 The Rules The Rules 1. To "undo" a mathematical operation, you must do the opposite. 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same to the other side. We learned earlier that for every mathematics operation, there is an inverse operation which undoes it: when you do both If the two expressions on the opposite sides of the equal sign are operations, you get back to where you started. equal to begin with, they will continue to be equal if you do the same mathematical operation to both of them. When the variable for which we are solving is connected to something else by a mathematical operation, we can eliminate This allows you to use an inverse operation on one side, to undo that connection by using the inverse of that operation. an operation, as long as you also do it on the other side. You can just never divide by zero (or by something which turns out to be zero) since the result of that is always undefined. Slide 29 / 138 Slide 30 / 138 Solving Equations The Rules 3. You can always switch the left and right sides of an equation. When solving equations, the goal is to isolate the variable on one side of the equation in order to determine its value (the value that makes the equation true). Once an equation has been solved for a variable, it'll be a lot easier to use if that variable is moved to the left side. x + 7 = 32 Mathematically, this has no effect since the both sides are equal. But, it's easier to use the equation if the side for which you are solving is on the left and values are substituted on the right. The goal: get "x" by itself on one side of the equal sign.
Slide 31 / 138 Slide 32 / 138 Inverse Operations 5 What is the inverse operation needed to solve this equation? For each equation, write the inverse operation needed to solve for the variable. 7x = 49 a.) y + 7 = 14 subtract 7 b.) a - 21 = 10 add 21 click click A Addition B Subtraction C Multiplication c.) 5s = 25 divide by 5 d.) x = 5 multiply by 12 D Division click click 12 Slide 33 / 138 Slide 34 / 138 6 What is the inverse operation needed to solve this 7 What is the inverse operation needed to solve this equation? equation? x - 3 = 12 x = 8 5 A A Addition Addition B Subtraction B Subtraction C Multiplication C Multiplication D Division D Division Slide 35 / 138 Slide 36 / 138 8 What is the inverse operation needed to solve this equation? 25 + x = 30 Solving One Step A Addition Addition & Subtraction B Subtraction Equations C Multiplication D Division Return to Table of Contents
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