Slide 1 / 258 Slide 2 / 258 7th Grade Equations & Inequalities 2015-12-08 www.njctl.org Slide 3 / 258 Slide 4 / 258 Table of Contents Click on a topic to Equations and Identities go to that section. Solving an Equation for a Variable One Step Equations Two Step Equations Equations and Identities Multi Step Equations Distributing Fractions in Equations Writing and Solving Algebraic Equations Graphing and Writing Inequalities in One Variable Simple Inequalities Involving Addition and Subtraction Simple Inequalities Involving Multiplication and Division Return to Table Glossary & Standards of Contents Slide 5 / 258 Slide 6 / 258 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 / 258 Slide 8 / 258 Equations and Identities Equations and Identities Here are some equations: Here are some identities: s = 15/t 2 + 3 = 5 v = 12 - 9.8t 9 - 2 = 7 x = 10 + 6t - 4.9t 2 5(x - 3) = 5x - 15 In all these cases, the variables are interdependent. They are 0.5y = y/2 only true for certain sets of variables. These are always true...there are no values that can be Changing the value of t on the right side of the equation, assigned to the variables for which these would be untrue. changes the possible values of s, v or x on the 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 / 258 Slide 10 / 258 Equations and Identities Equations and Identities s = 15/t s = d/t v = 12 - 9.8t v = v o + at x = 10 + 6t - 4.9t 2 x = x o + v o t + 1/2at 2 Above are simplified physics equations in which t In these cases, there are up to five variables, which all represents time, s represents the speed; v is for final depend on each other. velocity and x is for final position. You'll work with these more in Algebra I, but it's important to In these equations, we have only two variables, but in see that equations define the relationship between variables later math courses, there will be more than two. since the equation is only true for certain sets of values. For instance, here are more general versions of those same equations. Slide 11 / 258 Slide 12 / 258 Tables and Equations Tables and Equations Note that in the first three sets of answers, the object was Let's use this table to find some solutions to the equation s = d/t; moving at a speed of 15 m/s. where s represents speed (in meters/second), d represents distance (in meters) and t represents time (in seconds). The final three sets of answers are for an object traveling four times faster, at 60 m/s. s = d/t s = d/t We've entered the distance But, in all cases, knowing t (s) t (s) d (m) s (m/s) d (m) s (m/s) traveled and the time it took the value of two of the three 30 2 30 2 15 to travel that distance in two variables determines the of the columns. 60 4 values of the third. 60 4 15 90 6 90 6 15 Use the equation (s = d/t) to 120 2 120 2 60 find the speeds and fill in the blank column. 240 4 240 4 60 360 6 360 6 60
Slide 13 / 258 Slide 14 / 258 Tables and Equations Tables and Equations Knowing any two of the values, determines the third. This was more challenging, since the equation is solved for s...not for d or t. Try it in this case, where the equation is still s = d/t, but the values of different variables are provided. Math Practice Filling in values will be even more challenging when we work with equations that have more variables. Can you still figure out the unknown values and complete the table? s = d/t s = d/t We need a way to solve an d (m) t (s) s (m/s) equation for any variable, so d (m) t (s) s (m/s) 20 2 10 that we can find its value, 20 2 given the values of the others. 100 5 20 5 20 100 10 10 Solving an equation for a 100 10 variable means to get it alone 120 2 60 120 2 on the left side of the equation. 240 3 80 240 80 360 6 60 6 60 Slide 15 / 258 Slide 16 / 258 Solving for a Variable Our goal is to be able to solve any equation for any variable that appears in it. Let's look at a simple equation first. Solving an Equation d for a Variable 8 = 10 The variable in this equation is d. Solving for a variable means having it alone on the left side . Right now, the equation is not solved for a variable. Return to Table of Contents Slide 17 / 258 Slide 18 / 258 The Rules for Solving Equations The Rules Like in any game there are a few rules. Here are the four rules. Let's examine them, one at a time. There are four rules which will allow you to solve any equation for 1. To "undo" a mathematical operation, you must perform the any of its variables, not just in 7th grade Math, but in most any of inverse operation. your subsequent mathematics and science courses. 2. You can do anything you want (except divide by zero) to one So, learning these rules and how to apply them is very important. side of an equation, as long as you do the same thing to the other. 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite of the "order of operations." 4. You can always switch the left and right sides of an equation.
Slide 19 / 258 Slide 20 / 258 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 21 / 258 Slide 22 / 258 The Rules The Rules 3. If there is more than one operation going on, you must undo 4. You can always switch the left and right sides of an equation. them in the opposite order in which you would do them, the opposite of the "order of operations." 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. The operations which are connected to a variable must be "undone" in the reverse order from the Order of Operations. Mathematically, this has no effect since the both sides are equal. So, when solving for a variable, you: But, it's easier to use the equation if the side for which you are first have to do addition/subtraction, solving is on the left and values are substituted on the right. then multiplication/division, then exponents/roots, finally parentheses. The order of the steps you take to untie a knot are the reverse of the order used to tie it. Slide 23 / 258 Slide 24 / 258 The Rules 1 First, is d already alone? If not, what is with it? A 8 Let's solve this equation for "d" B d C 10 D d it is already alone 8 = 10 That means that when we're done d 8 = we'll have d alone 10 on the left side of the equation.
Slide 25 / 258 Slide 26 / 258 3 What is the opposite of dividing d by 10? 2 What mathematical operation connects d and 10? dividing 10 by d A A d is added to 10 B dividing by s into 10 B d is multiplied by 10 C multiplying d by 10 C d is divided by 10 multiplying by 10 by d D D 10 is subtracted from d d 8 = 10 d 8 = 10 Rule 1. To "undo" a mathematical operation, you must do the opposite. Slide 27 / 258 Slide 28 / 258 5 Is there more than one mathematical operation acting 4 What must we also do if we multiply the right side by 10? on "d"? A divide the left side by 10 Yes B multiply the left side by 10 No C divide the left side by d D divide the left side by d d 8 = 10 d 8 = 10 Rule 3. If there is more than one operation going on, you must undo Rule 2. You can do anything you want (except divide by zero) to them in the opposite order in which you would do them, the opposite one side of an equation, as long as you do the same thing to the of the "order of operations." other. Slide 29 / 258 Slide 30 / 258 Applying Rule 4 Applying Rules 1 and 2 Rule 4. You can always switch the left and right sides of an equation. 1. To "undo" a mathematical operation, you must do the opposite. d 8 = 80 = d 10 2. You can do anything you want (except divide by zero) to one d = 80 side of an equation, as long as you do the same thing to the other. So we undo d being divided by t, by multiplying both sides by t. We've now solved our equation for d. (10) d (10) (8) = 10 80 = d Are we done?
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