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Lesson 6.2.4 Solving Multi-Step Equations Remember These? Take a minute and simplify the following three expressions. 1. β3 π₯ + 4 2. 3. β4π β 3(π + 8) β2π β 3 + 4π + 8 Priorβ¦ ο― In previous chapters we learned how to simplify expressions using the distributive property and by combining the βlike termsβ. ο― So far in Chapter 6, we have been concentrating mainly on equations that can be solved in one step or in two steps like the equation 2x + 5 = 17. ο― In this lesson we will look at solving multi-step equations. As the name suggestsβ¦ there are steps you must do before you can solve the equation. Steps for Solving any Equation 1) Distribute if you can. 2) Then combine the like terms. 3) Now you have a more simplified equation. Solve it by undoing in reverse. 4) Check your answer. ο― To be successful with multi-step equationsβ¦ you will have to be able to analyze the equation and decide what steps must be taken to solve itβ¦ ο― Be on the look out for opportunities to distribute and to combine like terms when solving the equations today! Letβs Get Startedβ¦ Section 1: Combining Like Terms Section 2: Distributive Property Section 3: Multi-Steps with Fractions. Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. Step 4 Check Guided Practice #1 ππ β ππ β π = ππ Step 1 is not needed butβ¦ on the left side of the equation I notice that I have two like terms (ππ± and βππ±). Since the variables are alike I can combine them to get ππ±. ππ± β π = ππ After I combine the terms I have a 2-step equation. To solve this equationβ¦ Undo the subtract 1st. Then undo the multiplication. Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. Step 4 Check ππ± β π = ππ Since this equation has β 8, I will add 8 to both sides The 8βs on the left cancel out ππ± β π = ππ +π + π ππ± = ππ I am left with a 1-step equation 24 + 8 = 32 ππ± = ππ In this instance 4x means 4 times x. To undo the multiplication divide both sides by 4 The 4βs on the left cancel out leaving x ππ± = ππ π π π± = π The solution that makes the statement true is x = 8 32 οΈ 4 = 8 Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. ππ± β ππ± β π = ππ π(π) β π(π) β π = ππ ππ β ππ β π = ππ ππ = ππ Step 4 Check You Try #1 ππ± β ππ± β ππ = ππ Moving Onβ¦ Section 1: Combining Like Terms Section 2: Distributive Property Section 3: Multi-Steps with Fractions. Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. Step 4 Check Guided Practice #2 5x + 3(x +4) = 28 In this instance I begin on the left side of the equationβ¦ 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) that I have to combine them to get 8x before I can solve this equationβ¦ 8x + 12 = 28 Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. Step 4 Check 8x + 12 = 28 I am now left with a 2-step equation The left side has +12. To undo the +12, I subtract 12 from both sides The 12βs on the left cancel out leaving 8x 8x + 12 = 28 -12 -12 8x = 16 I am left with a 1-step equation 28 β 12 = 16 8x = 16 On the left side 8x means 8 times x. To undo the multiplication I divide both sides by 8 8x = 16 The 8βs on the left cancel out leaving x 8 8 16 οΈ 8 = 2 x=2 The solution that makes the statement true is x = 2 Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. Step 4 Check ππ± + π(π± + π) = ππ π π + π(π + π) = ππ ππ + π(π) = ππ ππ + ππ = ππ ππ = ππ You Try #2 π π± β π + ππ± = π Distributing a Negative ο― Distributing a negative number is similar to using the distributive property. ο― However, students get this wrong because they forget to use the rules of integers ο― Quickly the rules areβ¦when multiplying, if the signs are the same the answer is positive. If the signs are different the answer is negative Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. Step 4 Check Guided Practice #3 4x β 3(x β 2) = 21 I begin by working on the left side of the equationβ¦ 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 β 3(x β 2) = 21 4x β 3x + 6 = 21 After doing the distributive property, I see that I can combine the 4x and the -3x to get 1x or x : x + 6 = 21 Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. x + 6 Step 4 Check = 21 After combining those like terms, you are left with a simple one step equation. To undo the +6 subtract 6 from both sides of the equation 21 β 6 = 15 x The 6βs cancel out leaving x +6 - 6 = 21 -6 x = 15 The solution is x = 15 Step 1 Distribute Step 2 Combine like terms Step 3 Solve by undoing in reverse. Step 4 Check ππ β π(π β π) = ππ π ππ β π(ππ β π) = ππ ππ β π(ππ) = ππ ππ β ππ = ππ ππ = ππ You Try #3 ππ± β π π± β π = ππ Moving Onβ¦ Section 1: Combining Like Terms Section 2: Distributive Property Section 3: Multi-Steps with Fractions and Decimals. Review Step 1: Look at all of the denominators in the equation and find the LCD. Step 2: Multiply both sides of the equation by the LCD. π± π + =π π π The LCM is the smallest number that both 2 and 5 divide into evenly. LCD of 2,5 = ππ π± π ππ ( + ) = π (ππ) π π π π π π 10 ( ) + ππ ( ) = 10(3) ππ± + π = ππ Fractions are gone! Review Step 1: Step 2: Look for the decimal with the most digits. Multiply both sides of the equation by that power of 10. Decimal with the most digits. π. πππ + π. π = π. π πππ π. πππ + π. π = πππ π. π πππ π. πππ + πππ π. π = πππ(π. π) πππ + ππ = ππ Decimals are gone! Step 1 Distribute Step 3 Solve by undoing in reverse. Step 2 Combine like terms Step 4 Check Guided Practice #4 π π± + π = ππ ππ In this example, there are actually a couple of ways that you can correctly handle the fractionβ¦ However, we are going to keep with our steps and distribute first. Then we will clear fractions. π ππ π± + π = ππ π π± ππ Now we will clear these fractions. + π ππ = ππ Step 1 Distribute Step 2 Combine like terms 10 10 π π π± + ππ ππ Step 3 Solve by undoing in reverse. Step 4 Check 10 = ππ To clear the fractions, we multiply both sides of the equation by the LCDβ¦ which is 10. The result is an equation without fractions. ππ± + π = πππ There are no like terms to combine. All we need to do now is to solve the 2-step equation by undoing in reverse. Step 1 Distribute Subtract 6 from both sides Step 3 Solve by undoing in reverse. Step 2 Combine like terms Step 4 Check ππ± + π = πππ βπ βπ ππ± = πππ π π π± = ππ Then divide by 3. The solution to the equation is x = 38 Letβs move on to another practice problemβ¦ but donβt forget that you should always check your solution. Guided Practice #4 π ππ± + π = βππ π You Try #4 π π β π = βππ π Guided Practice #5 π. ππ π + π = ππ You Try #5 π. π π β π = βππ Always More Than One Way Always More Than One Way Summary ο― With multi-step equationsβ¦ you have to be able to analyze the equation and decide what steps must be taken to solve itβ¦ Summary 1) Distribute if you can. 2) Then combine the like terms. 3) Now you have a more simplified equation. Solve it by undoing in reverse. 4) Check your answer. These steps can be used to solve any equation. You will not always have to do Step 1 and Step 2. Summary There is always more than one way to solve a problem. In this equation, you can distribute first if you want to, butβ¦ we learned today that you can also divide both sides by 3.
