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One Step Equations – Addition and Subtraction

One Step Equations – Addition and Subtraction. ANSWER. X = -23. 1. Draw a vertical. and horizontal line. x. + 8. = -15. What’s the goal? Get x by itself. – 8. – 8. x. -23. What do you do?. Get rid of “ +8 ”. =. -8. How?. What’s the inverse of +8 ?.

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One Step Equations – Addition and Subtraction

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  1. One Step Equations – Addition and Subtraction ANSWER X = -23 1. Draw a vertical and horizontal line x + 8 = -15 What’s the goal? Get xby itself. – 8 – 8 x -23 What do you do? Get rid of “+8” = -8 How? What’s the inverse of +8 ? So, subtract 8from the left, …but what you do to one side… The 8’s on the variable side “cancel”. Bring down your x Add -15 and -8

  2. One Step Equations – Addition and Subtraction One Step Equations ? 13 17.5– 4.5 = EXAMPLE 2 2. c – 4.5 = 13 + 4.5 + 4.5 To get c by itself, add 4.5 to both sides. The 4.5’s on the variable side “cancel”. c = 17.5 Check c – 4.5 = 13 How can you check to make sure? Substitute 17.5 for c in original equation. ✓ Does the left side really equal 13? 13

  3. One Step Equations – Multiplication and Division y y -5 -5 = = 3 3 -15 = EXAMPLE 1 Goal? Get yby itself. What do you do? Get rid of “ ” 3 • 3 How? What’s the inverse of ? 3 3 Multiply each side by 3 . 3 The 3’s on the variable side “cancel”. y Check -15 ? Substitute -15 for y in original equation. ✓ Are both sides equal? -5

  4. One Step Equations – Multiplication and Division -20 -2.5x = -20 -2.5x = 8 = x To get x by itself, you have to get rid of “-2.5” What’s the inverse of multiplying by -2.5 dividing by -2.5 -2.5 -2.5 The -2.5’s on the variable side “cancel”. Check -2.5x =-20 Substitute 8for x ? -2.5 • 8 =-20 Are both sides equal? ✓ -20

  5. Solving a One Step Equation With Fractions Nothing changes -- cc. + a = = Do what you normally do to get a by itself. = Rename each fraction with a common denominator. a = Both signs are negative, so are we “adding” or “subtracting”? We’re adding, so add the numerators,. …and keep the sign negative. SOLUTION: a =

  6. Solving a One Step Equation With Fractions dd. ee. ff. gg. kk. hh. ii. jj.

  7. Solving a One Step Equation With Fractions To get xby itself, you must multiply by the “reciprocal of the coefficient”. Huh? • • The coefficient is , so the reciprocal is So, multiply both sides by On the left, the 3’s cancel . . . and the 2’s cancel On the right, multiply 10 • x = or x = 15. Solve for x.

  8. Solving a One Step Equation With Fractions 43. 44. • • - t = -12 t = or t = 30 Z = - 46. 45. x = -40 x = -16

  9. Solving a One Step Equation With Fractions

  10. Solving a Two Step Equation Look closely at this equation. To get n by itself, we’ll need to get rid of 2370 = 395n + 1580 – 1580 - 1580 1580 and 395 To start, subtract 1580from each side. 790 = 395n 395 395 Then, divide both sides by 395. 2 = n or n = 2

  11. Solving a Two Step Equation x x x – 14 8 = – – – 2 2 2 12 12 12 2x 2x 8 8 8 = = = -2 +14 +14 - 12 - 12 – – 4 2x = 2 = 22 2 = 22 44 ? –14 = 8 2 x 2 = EXAMPLE 2 1. 2. -2 or x = 2 x = 44 Check Check •2 4 22 – 14 = 8 8

  12. Solving a Two Step Equation y z – 10 4a 34 6 – = 16 – a – 55. 4 53. 6x 5 = 23 52. 13 = 11 + 50. 54. – 8w 3 = 21 3 51. = = + 3 5 6 = y or y = 6 10 = a or a = 10 z = 35 a = -6 w = 3 x = -3

  13. Writing and Solving a Two-Step Equation EXAMPLE 2 1. The sum of 4 times a number and –6 is 14. What is the number? and –6 is 14 The sum of 4 times a number 4x –6 = + 14 4n – 6 = 14 n = 5 ANSWER The number is 5. 2. The quotient of a number and twenty, decreased by five, is fifteen. The quotient of a number and twenty decreased by five is fifteen. – 5 = 15 ANSWER The number is 400.

  14. Writing and Solving a Two-Step Equation 1. Are you sure? 40.8 less than a number is negative eighty Think: What’s “seven less than ten”? - Isn’t that just “ 10 – 7 ” ? x = -80 40.8 + 40.8 So, every time you see the phrase “ less than “, you should… x = -39.2 x = -39.2 2. 20 is four less than 3 times a number. 20 = 3x - 4 The number is 8.

  15. Writing and Solving a Two-Step Equation Write an equation, then find the missing number. 1. The difference of six times a number and 9 is -3. ANSWER The number is 1. ANSWER 2. The sum of a number and 6.4 is negative 23. The number is 29.4. 3. The difference of twelve and the product of three and a number is twenty-four. ANSWER The number is -4.

  16. Writing and Solving a Two-Step Equation 10 10 You pay $35 in dues to a video game club and $10 per game you rent. If you spend $225 during the year, how many games did you rent? Write a formula, then solve for x. DUES + (# OF GAMES) • (GAME COST) = TOTAL COST Make a quick model Plug in what you know. 35+ ( ) • ( 10) = 225 g Since you don’t know the “# of games”, let’s call it g. 35 + 10g = 225 Multiply g •10and rewrite. To get 10gby itself, subtract 35. – 35 – 35 10g = 190 To get g by itself, divide by 10. ANSWER g = 19 Solve for g. You rented 19 games.

  17. Writing and Solving a Two-Step Equation 7.92 = ANSWER You downloaded 8 songs. You pay $9.95 to join an Internet music club. You pay $.99 for each song that you download. Your cost for joining and downloading some songs is $17.87. How many songs did you download? Write a model. 9.95 0.99x 0.99 x = 17.87 + • –9.95 –9.95 To get the x term by itself, subtract9.95from each side. 0.99x Divide each side by0.99 to undo multiplication. 0.99 0.99 x = 8 Solve for x.

  18. Writing and Solving a Two-Step Equation EXAMPLE 3 Reading You want to find how many pages you need to read per week in order to finish several books in six weeks. The books have a total of 1244 pages. You have already read 500 pages. How many pages do you have to read per week? Let pbe the number of pages to read per week. 1244 = 6p+ 500 124 = p ANSWER You have to read 124 pages per week.

  19. Writing and Solving a Two-Step Equation 2370 = 395 • n + 1580 or 2370 = 395n + 1580 – 1580 - 1580 790 = 395n Subtract 1580from each side. 395 395 2 = n Divide both sides by 395. The zoo can buy 2 new flamingos next year.

  20. Writing and Solving a Two-Step Equation GUIDED PRACTICE Feeding Costs It costs a zoo $1150 this year to feed tortoises. Each new tortoise costs $575 to feed. A zoo’s budget for tortoise food next year is $2875. How many new tortoises can the zoo buy next year? ANSWER The Zoo can buy 3 new tortoises next year

  21. Solving Multi-Step Equations –1 –1 EXAMPLE 2 3x +12 – 4x = 20 3x+12 – 4x = 20 Combine like terms. –x+ 12 = 20 - 12 - 12 Subtract 12 from each side. –x = 8 Hint: -x = -1x –1 x = 8 so rewrite it as -1x = 8 Divide each side by – 1. x = -8 Solve.

  22. Solving Multi-Step Equations 3. 4p +10 + p = 25 –8r – 2 +7r =– 9 2. –6 = 11w –5w 1. Solve the equation. Then check the solution. w = – 1 p = 3 r = 7

  23. Writing and Solving Multi-Step Equations = + 15c 75 = 15 15 You can plant five cups with 5 seeds and five cups with 10 seeds. ANSWER For a science fair, you perform an experiment to see how the number of Venus flytrap seeds planted in a cup affects plant growth. In each cup, you plant either 5 seeds or 10 seeds. You want to use an equal number of cups for each seed amount. You have 75 seeds. How many cups do you need? You can use an equation to solve this problem. 5c+ 10c= 75 15c = 75 Combine like terms. Divide each side by 15. c = 5

  24. Writing and Solving Multi-Step Equations ANSWER Flowers You buy 6 packs of each type of flower. A pack of petunia seeds costs $3 and a pack of balloon flower seeds costs $4. You buy the same number of packs of each type of flower and pay $42. How many packs of each do you buy? What are we trying to find? Since we’re trying to find the number of packs, let’s call it p. $3 • p+ $4 • p= 42 3p+ 4p= 42 7p= 42 Combine like terms. p= 6 Solve for p.

  25. Solving Multi-Step Equations EXAMPLE 3 6n –2(n +1) = 26 Use Distributive property 6n –2(n +1) = 26 “outer times first”, then 6n –2n –2 = 26 “outer times second”, 6n – 2n – 2 = 26 6n– 2n – 2 = 26 Combine like terms. 4n – 2 = 26 + 2 + 2 Add 2to each side. 4n = 28 Solve. n = 7

  26. Solving Multi-Step Equations 1. 2. 3. 3(x – 9) = – 39 -63 = –7(8 – p) 25 = -3(2x + 1) p = -1 x = - 4 x = – 4

  27. Solving Multi-Step Equations 55 + 3x = 8x 114. GUIDED PRACTICE Solve the equation. What’s the goal? Get the variables on one side... …and the constants on the other. …so, if you get rid of 3x on the left, you’ll have it. - 3x - 3x 55 = 5x Solve. 11 = x or x = 11

  28. Solving Multi-Step Equations 9x = 12x – 9 115. 116. –15x + 120 = 15x GUIDED PRACTICE x = 3 4 = x

  29. Solving Multi-Step Equations 117. 4a + 5 = a + 11 GUIDED PRACTICE Get the variables on one side... …and the constants on the other. …but, which side for each? It doesn’t really matter. -a -a Hint: Move the smaller variable to the larger variable’s side. 3a + 5 = + 11 - 5 - 5 Subtract 5to isolate the variable. 3a = 6 Solve. a = 2

  30. Solving Multi-Step Equations 118. 3n + 7 = 2n –1 119. –6c + 1 = –9c + 7 n = –8 c = 2 120. 11 + 3x – 7 = 6x + 5 – 3x 121. 6x + 5 – 2x = 4 + 4x + 1 there are no solutions for x all values of x are solutions

  31. Solving Multi-Step Equations GUIDED PRACTICE 122. 4(w – 9) = 7w + 18 123. 2(y + 4) = –3y – 7 y = –3 w = –18

  32. Writing and Solving Multi-Step Equations Let xrepresent the price of one tube of wax. 2x 15.00 = ANSWER 2 2 The price of one tube of wax is $7.50. 124. You and a friend are buying snowboarding gear. You buy a pair of goggles that costs $39.95 and 4 tubes of wax. Your friend buys a helmet that costs $54.95 and 2 tubes of wax. You each spend the same amount. Write and solve an equation to find the price of one tube of wax. 39.95 + 4x= 54.95 + 2x Write an equation. 39.95 + 2x= 54.95 Subtract 2xfrom each side. 2x = 15.00 Subtract 39.95 from each side. Divide each side by 2 x = 7.50 Solve.

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