Solving Multi-Step Equations Lesson 2.3 0011 0010 1010 1101 0001 0100 1011 1 2 4 California 0011 0010 1010 1101 0001 0100 1011 Standards 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12. 5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. 1 2 4 Solving Multi-step Equation • Steps – 0011 0010 1010 1101 0001 0100 1011 1. Circle like terms. 2. Combine like terms. 3. Isolate Variable 1. Add/subtract 2. Multiply/divide 3. Solve 2x + 3x – 4 =11 5x – 4 = 11 +4 +4 1 5x = 15 5 5 x=3 2 4 More Example Solve the equation. Check your answer. 0011 0010 1010 1101 0001 0100 1011 1.undo the division. 2. undo the addition. 2x + 1 = 21 –1 –1 2x = 20 x = 10 3. undo the multiplication. 4. The solution set is {10}. 1 2 4 Now let check our answer Solve the1010 equation. 0011 0010 1101Check 0001 your 0100answer. 1011 Check To check your solution, substitute 10 for x in the original equation. 1 7 7 2 4 Let Try again Solve 4 = 2x + 5 – 6x 4 = 2x + 5 – 6x 0011 0010 1010 1101 0001 0100 1011 4 = 2x – 6x + 5 4 = –4x + 5 –5 –5 –1 = –4x Combine like terms. undo the addition. undo the multiplication. 1 2 4 The solution set is . Try some on your own. 1. 46 = 4x – 4 + 6x 4. 0011 0010 1010 1101 0001 0100 1011 5=x 2. 4x – 8 + 2x = 40 x= 8 3. 36 = 10a – 12 – 7a 16 = a 5. 1 x=6 6. –8 – 2d + 2 = 4 2 4 d = -5 You may have to combine like terms or use the Distributive Property before you begin solving. 0011 0010 1010 1101 0001 0100 1011 1 2 4 Using Distributive Property Solve the equation. 0011 0010 1010 1101 0001 0100 1011 5(p – 2) = –15 5(p – 2) = –15 5(p) + 5(–2) = –15 5p – 10 = –15 +10 +10 5p = –5 Distribute 5. Simplify. 1 2 Since 10 is subtracted from 5p, add 10 to both sides. 4 Since p is multiplied by 5, divide both sides by 5. p = –1 The solution set is {–1}. 0011 0010 1010 1101 0001 0100 1011 Helpful Hint You can think of a negative sign as a coefficient of –1. –(x + 2) = –1(x + 2) and –x = –1x. 1 2 4 Another Example Solve the equation. 0011 0010 1010 1101 0001 0100 1011 10y – (4y + 8) = –20 10y +(–1)(4y + 8) = –20 Write subtraction as the addition of the opposite. 10y + (–1)(4y) + (–1)(8) = –20 Distribute –1. 10y – 4y – 8 = –20 6y – 8 = –20 +8 6y +8 = –12 y= -2 Simplify. 1 2 4 Combine like terms. Since 8 is subtracted from 6y, add 8 to both sides to undo the subtraction. Try some on your own. 1. –4(2 – y) = 8 y = 4 4. d + 3(d – 4) = 20 0011 0010 1010 1101 0001 0100 1011 2. 3(x + 1) – 4 = 5 d=8 X=2 3. x – (12 – x) = 38 25 1 2 4 Application Lin sold 4 more shirts than Greg. Fran sold 3 times as 0011 0010 1010 1101 0001 0100 1011 many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell? Locate key words in the question. UNDERLINE WHAT YOU ARE BEING ASKED TO FIND 1 2 Reread and circle relevant information 1 2 4 Lin sold 4 more shirts than Greg. Fran sold 3 times as 0011 0010 1010 1101 0001 0100 1011 many shirts as Lin. In total, the three sold 51 shirts. How many shirts did Greg sell? 3 Reread the part of the problem you underlined, and define an appropriate variable (or variables) 1 2 Since the information is given in relation to Lin, set an equation for each individual in terms of Lin. Greg = L – 4 Lin = L Fran = 3L 4 Lin sold 4 more shirts than Greg. Fran sold 3 times as many shirts as Lin. In total, the three sold 51 shirts. 0011 0010 1101 0001 did 0100Greg 1011 sell? How1010 many shirts Write an equation (or inequality) and then check to see if it’s correct by rereading the circled and (L – 4) + (L) + underlined (3L) = 51information Substitute. –4= 51 Combine like Greg Greg + Lin5L + Fran = 51 terms. = L – 4 +4 +4 Lin = L Since 4 is subtracted from 5L add 4 to 5L = 55 = the 3Lsubtraction. both sidesFran to undo 4 1 L = 11 2 4 Since L is multiplied by 5, divide both sides by 5 to undo the multiplication. More Application At a local gym, there is a joining fee of 0011 0010 1010 1101 0001a0100 1011 $59.95 and monthly membership fee. Sara and Martin both joined this gym. Their combined cost for 12 months was $1319.90. How much is the monthly fee? 1 Step 1: Step 3: 4: 2: 2 Let m represent monthly feefee paidisby each. initial total 12 theplus Monthly cost. for 2 months fee for 2 2 (12m + 119.90) 4 = 1319.90 2(12m + 59.95) = 1319.90 0011 0010 1010+1101 0001 0100 2(12m) 2(59.95) =1011 1319.90 Distribute 2. 24m + 119.90 = 1319.90 –119.90 –119.90 Since 119.90 is added to 24m, subtract 119.90 from both 24m = 1200.00 sides to undo the addition. 1 2 Since m is multiplied by 24, divide both sides by 24 to undo the multiplication. 4 m = 50 Sara and Martin each paid $50 per month. Lesson Quiz: Part l Solve each 0011 0010 1010 1101equation. 0001 0100 1011 1. 2y + 29 – 8y = 5 2. 3(x – 9) = 30 19 3. x – (12 – x) = 38 4. 4 9 25 1 2 4 5. If 3b – (6 – b) = –22, find the value of 7b.–28 Lesson Quiz: Part ll 0011 0010 1010 1101 0001 0100 1011 6. Josie bought 4 cases of sports drinks for an upcoming meet. After talking to her coach, she bought 3 more cases and spent an additional $6.95 on other items. Her receipts totaled $74.15. Write and solve an equation to find how much each case of sports drinks cost. 1 2 4 4c + 3c + 6.95 = 74.15; $9.60