Slide 1 / 188 Slide 2 / 188 8th Grade Math Solving Equations 2015-12-17 www.njctl.org Slide 3 / 188 Slide 4 / 188 Table of Contents Review of Two Step Equations Click on a topic to go to that section. Multi-Step Equations Solving Equations that Contain Fractions Review of Two-Step Equations with the Same Variable on Both Sides Equations Comparing Expressions with the Same Variable Writing and Solving Algebraic Equations Translating & Solving Consecutive Integer Problems Glossary & Standards Return to Table of Contents Slide 5 / 188 Slide 6 / 188 Two-Step Equations Tips for Solving Equations 1. To "undo" a mathematical operation, you must perform the A two-step equation is an equation that contains two inverse operation . operations. For example, it could contain multiplication and subtraction, like the equation below. 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 5x - 9 = 16 other. Or it could contain addition and division like this equation. 3. If there is more than one operation going on, you must undo x them in the opposite order in which you would do them, the + 11 = -6 4 opposite of the "order of operations." Before we start solving two-step equations, let's review some 4. You can always switch the left and right sides of an equation. tips for solving them.
Slide 7 / 188 Slide 8 / 188 Tips Explained Tips Explained 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. 1. To "undo" a mathematical operation, you must do the opposite. If the two expressions on the opposite sides of the equal sign are We learned earlier that for every mathematics operation, there is equal to begin with, they will continue to be equal if you do the an inverse operation which undoes it: when you do both same mathematical operation to both of them. operations, you get back to where you started. This allows you to use an inverse operation on one side, to undo When the variable for which we are solving is connected to an operation, as long as you also do it on the other side. something else by a mathematical operation, we can eliminate that connection by using the inverse of that operation. 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 9 / 188 Slide 10 / 188 Tips Explained Tips Explained 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 4. You can always switch the left and right sides of an equation. opposite of the "order of operations." Once an equation has been solved for a variable , it is typically The operations which are connected to a variable must be easier to use if that variable is moved to the left side. "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: first have to undo addition/subtraction, 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 11 / 188 Slide 12 / 188 1 Is y already alone? If not, what is with it? Select all Solving for y that apply. Let's solve this equation for "y" A -4 B y -4y - 11 = -27 C -11 D -27 That means that when we're done we'll have y alone on the left side of the equation. E it is already alone -4y - 11 = -27
Slide 13 / 188 Slide 14 / 188 2 Which math operations connect the numbers to y? 3 Which math operation gets undone first? Select all that apply. A Addition A Addition B Subtraction C Multiplication B Subtraction Division D C Multiplication D Division -4y - 11 = -27 -4y - 11 = -27 Slide 15 / 188 Slide 16 / 188 4 What must we do if we add 11 to the left side? Solving for y 1. To "undo" a mathematical operation, you must do the opposite. A Subtract 11 from the left side -4y - 11 = -27 B Subtract 11 from the right side 2. You can do anything you want (except divide by zero) to one C Add 11 to the left side side of an equation, as long as you do the same thing to the other. So we undo 9 being subtracted from -2y by adding 9 to both D Add 11 to the right side sides. -4y - 11 = -27 + 11 +11 -4y = -16 Are we done? -4y - 11 = -27 Slide 17 / 188 Slide 18 / 188 5 What math operation connects -4 and y? 6 What is the opposite of multiplying y by -4? A -4 is being added to y A Dividing y by 4 B -4 is being subtracted by y B Dividing y by -4 C -4 is being multiplied by y C Multiplying y by 4 D -4 is being divided by y D Multiplying y by -4 -4y = -16 -4y = -16
Slide 19 / 188 Slide 20 / 188 7 Is y alone on the left? If not, what is with it? Solving for y A -11 1. To "undo" a mathematical operation, you must do the opposite. B -4 C y -4y = -16 D it is alone 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. y = 4 -4y = -16 -4 -4 y = 4 Slide 21 / 188 Slide 22 / 188 Solving for t 8 Is t already alone? If not, what is with it? Let's solve this equation for "t" A 5 B 15 15 C t 5 = t D it is already alone That means that when we're done we'll have t alone on the left side of the equation. 15 5 = t Slide 23 / 188 Slide 24 / 188 10 What is the opposite of dividing 15 by t? 9 What mathematical operation connects d to t? A dividing 15 by t A t is being divided by 15 B dividing 5 by t B 15 is being divided by t C multiplying 15 by t C 15 is being multiplied by t D multiplying t by 15 D t is being subtracted from 15 15 5 = 15 t 5 = t Rule 1. To "undo" a mathematical operation, you must do the opposite.
Slide 25 / 188 Slide 26 / 188 12 Is there more than one mathematical operation acting 11 What must we do if we multiply the right side by t? on "t"? A divide the left side by t Yes B multiply the left side by t No C divide the left side by 15 D divide the left side by 5 15 5 = 15 5 = t t Rule 2. You can do anything you want (except divide by zero) to Rule 3. If there is more than one operation going on, you must undo one side of an equation, as long as you do the same thing to the them in the opposite order in which you would do them, the opposite other. of the "order of operations." Slide 27 / 188 Slide 28 / 188 13 What mathematical operation connects 5 to t? Solving for t A t is being divided by 15 1. To "undo" a mathematical operation, you must do the opposite. B t is being divided into 5 C 15 t is being multiplied by 5 5 = t D t is being subtracted from 5 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. So we undo d being divided by t, by multiplying both sides by t. (t) 15 5t = 15 (t) 5 = t 5t = 15 Are we done? Slide 29 / 188 Slide 30 / 188 14 What is the opposite of multiplying t by 5? Solving for t A dividing t by 5 1. To "undo" a mathematical operation, you must do the opposite. B dividing t by t C multiplying t by t 5t = 15 D multiplying t by 5 5 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 5t = 15 other. 5t 15 = 5 5 t = 3
Slide 31 / 188 Slide 32 / 188 15 Is t alone on the left? If not, what is with it? Review of Solving Two-Step Equations A 5 B 15 The following formative assessment questions are C t review from 7th grade. If further instruction is need, see the presentation at: D it is alone https://www.njctl.org/courses/math/7th-grade/ equations-inequalities-7th-grade/ t = 3 Slide 33 / 188 Slide 34 / 188 16 Solve the equation. 17 Solve the equation. 14 = 3c + 2 5x - 6 = -56 Slide 35 / 188 Slide 36 / 188 18 Solve the equation. 19 Solve the equation. x 5r - 2 = -12 - 4 = 24 5
Slide 37 / 188 Slide 38 / 188 20 Solve the equation. 21 Solve the equation. x 14 = -2n - 6 + 7 = 13 5 Slide 39 / 188 Slide 40 / 188 22 Solve the equation. 23 Solve the equation. -2.5x - 4 = 3.5 x - + 2 = -10 3 Slide 41 / 188 Slide 42 / 188 24 Solve the equation. 25 Solve the equation. 3.3x - 4 = -13.9 -x + (-5.1) = -2.3 6
Slide 43 / 188 Slide 44 / 188 26 Solve the equation. 2.8x - 7 = -1.4 Multi-Step Equations Return to Table of Contents Slide 45 / 188 Slide 46 / 188 Steps for Solving Multiple Step Equations Multiple Step Equations As equations become more complex, you should: Example: 1. Simplify each side of the equation. (Combine like terms and use the distributive property.) 12h - 10h + 7 = 25 2. Use inverse operations to solve the equation. Remember, whatever you do to one side of an equation, you MUST do to the other side! Slide 47 / 188 Slide 48 / 188 Multiple Step Equations Hint Always check to see that both sides of the equation are Example: simplified before you begin solving the equation. 17 - 9f + 6 = 140 When an equation is simplified there should be at most one term for each variable and one constant term.
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