Multi-Step Equations This lesson will look at: Solving linear equations with 2 operations Combining like terms first Using the distributive property Distributing a negative Multiplying by a reciprocal first Your ability to be organized and lay out your problem will enable you to be successful with these concepts

Linear Equations with 2 Functions Sometimes linear equations will have more than one operation In this instance operations are defined as add, subtract, multiply or divide The rule is First you add or subtract Second you multiply or divide.

Linear Equations with 2 Operations You will have to analyze the equation first to see what it is saying… Then, based upon the rules, undo each of the operations… The best way to explain this is by looking at an example….

Example # 1 2x + 6 = 16 In this case the problem says 2 times a number plus 6 is equal to 16 Times means to multiply and plus means to add The first step is to add or subtract. To undo the addition of 6 I have to subtract 6 from both sides which looks like this: 16 minus 6 equals 10 The 6’s on the left side cancel each other out 2x + 6 = 16 - 6 -6 2x = 10 What is left is a one step equation 2x = 10

Example #1 (Continued) 2x = 10 The second step is to multiply or divide To undo the multiplication here you would divide both sides by 2 which looks like this: 10 divided by 2 is equal to 5 2x = 10 The 2’s on the left cancel out leaving x 2 2 x = 5 The solution to the equation 2x + 6 = 16 is x = 5

Combining Like Terms First When solving multi-step equations, sometimes you have to combine like terms first. The rule for combining like terms is that the terms must have the same variable and the same exponent. Example: You can combine x + 5x to get 6x You cannot combine x + 2x2 because the terms do not have the same exponent

Example # 2 I begin working on the left side of the equation 7x – 3x – 8 = 24 I begin working on the left side of the equation On the left side I notice that I have two like terms (7x, -3x) since the terms are alike I can combine them to get 4x. 7x – 3x – 8 = 24 4x – 8 = 24 After I combine the terms I have a 2-step equation. To solve this equation add/subtract 1st and then multiply/divide

Example # 2 (continued) Step 1: Add/Subtract Since this equation has – 8, I will add 8 to both sides 4x – 8 = 24 +8 +8 24 + 8 = 32 The 8’s on the left cancel out 4x = 32 I am left with a 1-step equation

The solution that makes the statement true is x = 8 Example # 2 (continued) 4x = 32 Step 2: Multiply/Divide In this instance 4x means 4 times x. To undo the multiplication divide both sides by 4 32  4 = 8 The 4’s on the left cancel out leaving x 4x = 32 4 4 x = 8 The solution that makes the statement true is x = 8

Solving equations using the Distributive Property When solving equations, sometimes you will need to use the distributive property first. At this level you are required to be able to recognize and know how to use the distributive property Essentially, you multiply what’s on the outside of the parenthesis with EACH term on the inside of the parenthesis Let’s see what that looks like…

Example #3 In this instance I begin on the left side of the equation 5x + 3(x +4) = 28 In this instance I begin on the left side of the equation I recognize the distributive property as 3(x +4). I must simplify that before I can do anything else 5x + 3(x +4) = 28 5x +3x +12 = 28 After I do the distributive property I see that I have like terms (5x and 3x) I have to combine them to get 8x before I can solve this equation

I am now left with a 2-step equation Example # 3(continued) 8x + 12 = 28 I am now left with a 2-step equation Step 1: Add/Subtract The left side has +12. To undo the +12, I subtract 12 from both sides 8x + 12 = 28 28 – 12 = 16 The 12’s on the left cancel out leaving 8x -12 -12 8x = 16

Example # 3 (continued) Step 2: Multiply/Divide On the left side 8x means 8 times x. To undo the multiplication I divide both sides by 8 8x = 16 16  8 = 2 The 8’s on the left cancel out leaving x 8 8 x = 2 The solution that makes the statement true is x = 2

Distributing a Negative Distributing a negative number is similar to using the distributive property. However, students get this wrong because they forget to use the rules of integers Quickly the rules are…when multiplying, if the signs are the same the answer is positive. If the signs are different the answer is negative

Example #4 4x – 3(x – 2) = 21 I begin by working on the left side of the equation. In this problem I have to use the distributive property. However, the 3 in front of the parenthesis is a negative 3. When multiplying here, multiply the -3 by both terms within the parenthesis. Use the rules of integers 4x – 3(x – 2) = 21 4x – 3x + 6 = 21 After doing the distributive property, I see that I can combine the 4x and the -3x to get 1x or x

The 6’s cancel out leaving x Example # 4 (continued) 4x – 3x + 6 = 21 x + 6 = 21 After combining like terms you are left with a simple one step equation. To undo the +6 subtract 6 from both sides of the equation x + 6 = 21 21 – 6 = 15 The 6’s cancel out leaving x -6 -6 x = 15 The solution is x = 15

Multiplying by a Reciprocal First Sometimes when doing the distributive property involving fractions you can multiply by the reciprocal first. Recall that the reciprocal is the inverse of the fraction and when multiplied their product is equal to 1. The thing about using the reciprocal is that you have to multiply both sides of the equation by the reciprocal. Let’s see what that looks like…

Example # 5 In this example you could distribute the 3/10 to the x and the 2. The quicker way to handle this is to use the reciprocal of 3/10 which is 10/3 and multiply both sides of the equation by 10/3 On the left side of the equation, after multiplying by the reciprocal 10/3 you are left with 120/3 which can be simplified to 40 On the right side of the equation the reciprocals cancel each other out leaving x + 2 The new equation is: 40 = x + 2

Example #5 (continued) 40 = x + 2 After using the reciprocals you are left with a simple one-step equation To solve this equation begin by working on the right side and subtract 2 from both sides of the equation The 2’s on the right side cancel out leaving x 40 = x + 2 40 – 2 = 38 - 2 -2 38 = x The solution to the equation is x = 38

Comments On the next couple of slides are some practice problems…The answers are on the last slide… Do the practice and then check your answers…If you do not get the same answer you must question what you did…go back and problem solve to find the error… If you cannot find the error bring your work to me and I will help…

Your Turn 6x – 4(9 –x) = 106 2x + 7 = 15 6 = 14 – 2x 3(x – 2) = 18

Your Turn 6. 7. 8. 9. 10. 5m – (4m – 1) = -12 55x – 3(9x + 12) = -64

Your Turn Solutions 14.2 4 8 1 ½ 3 -52 -13 -1 5