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Chapter 7 Notes Writing Algebraic Expressions Refer to operation word bank handout from earlier. Read statement, identify key words that identify the operation. Translate numbers and operations into algebraic expression. Translate into an algebraic expression Five more than triple a number. Six less than half a number. Four times the sum of a number and eight. Translate into an algebraic expression Five more than triple a number. 5 + 3n Six less than half a number. ½n–6 Four times the sum of a number and eight. 4(n + 8) Translate into an algebraic expression The price for six roses plus a delivery fee of $15 if roses are sold individually. The total cost for text messages if you are charged $0.75 per message. Translate into an algebraic expression The price for six roses plus a delivery fee of $15 if roses are sold individually. 6r + 15 The total cost for text messages if you are charged $0.75 per message. 0.75m Translate into an algebraic expression You have $30 to spend. You buy “b” books at $7 each. What is the cost for the books? How much change will you receive? Translate into an algebraic expression You have $30 to spend. You buy “b” books at $7 each. What is the cost for the books? 7b How much change will you receive? 30 – 7b Translate into an algebraic expression Mrs. Mathews is “x” years old. Mrs. Gibbs is eight more than half as old as Mrs. Mathews. Coach Van is 3 years younger than Mrs. Gibbs. Write an expression for Mrs. Gibbs age. Write an expression for Coach Van’s age. Write an expression that represents the sum of all their ages. Translate into an algebraic expression Mrs. Mathews is “x” years old. Mrs. Gibbs is eight more than half as old as Mrs. Mathews. Coach Van is 3 years younger than Mrs. Gibbs. Write an expression for Mrs. Gibbs age. ½ x + 8 Write an expression for Coach Van’s age. ½ x + 8 – 3 = ½ x + 5 Write an expression that represents the sum of all their ages. x+½x+8+½x+5 2x + 13 Translate into an algebraic expression The cafeteria sells pizza slices for $2 and chocolate chip cookies for $0.50. Write an expression to buy “p” slices of pizza and “c” cookies. How much change would you get from $20? Translate into an algebraic expression The cafeteria sells pizza slices for $2 and chocolate chip cookies for $0.50. Write an expression to buy “p” slices of pizza and “c” cookies. 2p + 0.50c How much change would you get from $20? 20 – (2p + 0.50c) 7-1 Solving 2-Step Equations To solve a 2-step equation, you will use the properties of equality. (whatever you do to one side you must do to the other.) Steps: first undo addition or subtraction. Then undo multiplication or division. Examples 3n - 6 = 15 +6 +6 3n = 21 3 3 N=7 Examples 15x + 3 = 48 R/4 - 10 = (-6) Examples - answers 15x + 3 = 48 -3 -3 15x = 45 15 15 x=3 r/4 - 10 = (-6) +10 + 10 r/4 = 4 4( r/4 ) = 4 (4) r = 16 Examples b/3 + 13 = 11 9g + 11 = 2 Examples - answers b/3 + 13 = 11 -13 -13 b/3 = -2 3(b/3) = 3(-2) b = -6 9g + 11 = 2 - 11 - 11 9g = -9 9 9 g = -1 Negative Coefficients Examples: 5 - x = 17 -a + 6 = 8 Negative Coefficients Examples: 5 - x = 17 -5 -5 -x = 12 -1 -1 X = -12 -a + 6 = 8 - 6 -6 -a = 2 -1 -1 a = -2 More Negative coefficients -9 - y/7 = (-12) 13 - 6f = 31 More Negative coefficients -9 - y/7 = (-12) +9 +9 -y/7 = -3 y/(-7) = -3 (-7) (y/(-7)) = (-3)(-7) Y= 21 13 - 6f = 31 -13 -13 -6f = 18 -6 -6 f = -3 Word Problems Lynne wants to save $900 to go to Puerto Rico. She saves $45 each week and now has $180. To find how many more weeks w it will take to have $900, solve 180 + 45w = 900. Word Problems -Answers Lynne wants to save $900 to go to Puerto Rico. She saves $45 each week and now has $180. To find how many more weeks w it will take to have $900, solve 180 + 45w = 900. 180 + 45w = 900 -180 - 180 45w = 720 45 45 w=16 It will take 16 weeks. Writing Equations Read the problem. Determine what you need to find. Define the variable you intend to use for the unknown. Write the equation then solve. Answer the problem using words and units. Check your solution for accuracy and reasonableness. Word Problem 1 Mike has $23. This is seven more than twice what Julie has. How much money does Julie have? Word Problem 1 Mike has $23. This is seven more than twice what Julie has. How much money does Julie have Let J = Julie 2j + 7 = 23 -7 -7 2j = 16 2 2 J = 8 Julie has $8 Word Problem 2 Brianna ran 45 miles over three day (Tues-Thurs). This is five less than four times the miles she ran Monday. How many miles did Brianna run on Monday? Word Problem 2 Brianna ran 45 miles over three day (Tues-Thurs). This is five less than four times the miles she ran Monday. How many miles did Brianna run on Monday? Let m = miles 4m – 5 = 45 + 5 +5 4m = 50 4 4 M = 12.5 Brianna ran 12.5 miles Word Problem 3 The temperature in Chicago was -7 degrees on Friday. Two weeks ago the temperature was 8 degrees more than half Friday’s temperature. That was the temperature two weeks ago? Word Problem 3 The temperature in Chicago was -7 degrees on Friday. Two weeks ago the temperature was 8 degrees more than half Friday’s temperature. That was the temperature two weeks ago? Let T=temperature ½ T + 8 = -7 -8 -8 ½ T = - 15 2( ½ T) = 2(-15) T = -30 The temp. was -30 degrees. 7-2 Solving Multi-Step Equations Combine like terms to simplify an equation before you solve it. Then solve -- undo addition or subtraction. Then multiply or divide. Combining Like Terms M + 2M - 4 = 14 3M - 4 = 14 +4 +4 3M = 18 3 3 M = 6 Example: 7 – y + 5y = 9 Finding Consecutive Integers Consecutive integers = when you count by 1’s from any integer (ex. 120, 121, 122, 123) Example: The sum of 3 consecutive integers is 96 N + (N+1) + (N+2) = 96 Using the Distributive Property 2(5x - 3) = 14 38 = (-3)(4y + 2) + y -3(m - 6) = 4 3(x + 12) - x = 8 Using the Distributive Property - Answers 2(5x - 3) = 14 X=2 38 = (-3)(4y + 2) + y y= -4 -3(m - 6) = 4 m= 4 2/3 3(x + 12) - x = 8 x= -14 7-3 Multi-Step Equations with Fractions and Decimals When there is a fraction next to a variable, you can do the reciprocal to solve the equation Examples 2 n - 6 = 22 3 -(7/10)k + 14 = (-21) 2/3(m - 6) = 3 Examples - Answers 2 n -6 = 22 3 n= 42 -(7/10)k + 14 = (-21) k=50 2/3(m - 6) = 3 m=10 1/2 Word Problems Suppose your cell phone plan has $20 per month plus $0.15 per minute. Your bill is $37.25. Use the equation 20 + 0.15x = 37.25. How many minutes are on your bill? Word Problems Answers Suppose your cell phone plan has $20 per month plus $0.15 per minute. Your bill is $37.25. Use the equation 20 + 0.15x = 37.25. How many minutes are on your bill? x=115 Consecutive Integers When you count by 1’s from any integer, you are counting consecutive integers Example: 45, 46, 47 When you count by 2’s from any number you are counting either consecutive odd or even integers Example: 2, 4, 6 or 3, 5, 7 Finding Consecutive Integers The sum of 3 consecutive integers is 96. Find the numbers. Find two consecutive even integers with a sum of 66. Find 2 consecutive even integers such that the sum of the larger and twice the smaller is 38. Finding Consecutive Integers - Answers The sum of 3 consecutive integers is 96. Find the numbers. n + (n+1) + (n+2)=96; 31, 32, 33 Find two consecutive even integers with a sum of 66. n + (n+2) = 66; 32,34 Find 2 consecutive even integers such that the sum of the larger and twice the smaller is 38. 2n + (n +2) = 38; 12, 14 7-4 Write an Equation Five times a number decreases by 11 is 9. Find the number such that three times the number increased by 7 is 52. Find a number such that seven less than twice the number is 43. 7-4 Write an Equation Answers Five times a number decreases by 11 is 9. 5n - 11 = 19 n = 6 Find the number such that three times the number increased by 7 is 52. 3n + 7 = 52 n = 15 Find a number such that seven less than twice the number is 43. 2n - 7 = 43 n = 25 Fifteen more than the product of 8 and a number is -17. Negative three times a number less four is 17. The product of 5 and a number increased by 10 is 145. Answers Fifteen more than the product of 8 and a number is -17. 15 + 8n = -17 n = -4 Negative three times a number less four is 17. -3n - 4 = 17 n = -7 The product of 5 and a number increased by 10 is 145. 5n + 10 = 145 n= 27 The difference between half a number and 9 is -23. The quotient of a number and 5, diminished by 11 is 18. Answers The difference between half a number and 9 is -23. 1/2n - 9 = -23 n = -28 The quotient of a number and 5, diminished by 11 is 18. N/5 - 11 = 18 n = 145 Rachel hung 38 ornaments on the tree. This is 3 less than half what Jane hung on the tree. How many ornaments did Jane hang on the tree? Sue did three more than twice the amount of sit-ups that Lisa did. If Sue did 67 sit-ups, how many did Lisa do? Answers Rachel hung 38 ornaments on the tree. This is 3 less than half what Jane hung on the tree. How many ornaments did Jane hang on the tree? 1/2n - 3 = 38 n = 82 Sue did three more than twice the amount of sit-ups that Lisa did. If Sue did 67 sit-ups, how many did Lisa do? 3 + 2n = 67 n = 32 Four friends go to dinner together. The check totals $36. They have a coupon for $4 off the total bill. They decide to split the check equally. How much does each person pay? Answers Four friends go to dinner together. The check totals $36. They have a coupon for $4 off the total bill. They decide to split the check equally. How much does each person pay? 4x +4 = 36 x = $8 each Mrs. Mathews has 3 more than twice the number of Christmas pins that Ms. Holden has. If Mrs. Mathews has 39 pins, how many does Ms. Holden have? The price of regular set of golf clubs was $179.95. The sale price was $113.25. How much do you save? Answers Mrs. Mathews has 3 more than twice the number of Christmas pins that Ms. Holden has. If Mrs. Mathews has 39 pins, how many does Ms. Holden have? 2x +3 = 39 x = 18 pins The price of regular set of golf clubs was $179.95. The sale price was $113.25. How much do you save? 113.25 + x = 179.95 x = $66.70 Word Problems Two-thirds the number of girls plus two represents the number of boys in the class. If there are 13 boys in the class, how many girls are there? Word Problems Answers Two-thirds the number of girls plus two represents the number of boys in the class. If there are 13 boys in the class, how many girls are there? 2/3y + 2 = 13 y = 16.5 7-5 Solving Equations with Variables on Both Sides To solve an equation with a variable on both sides, use addition or subtraction to collect the variable on one side of the equation. Collecting the variable on one side 9a + 2 = 4a - 18 4x + 4 = 2x + 36 k + 9 = 6(k - 11) Collecting the variable on one side - Answers 9a + 2 = 4a - 18 a = -4 4x + 4 = 2x + 36 x = 16 k + 9 = 6(k - 11) k = 15 Word Problem Beth leaves home on her bicycle, riding at a steady rate of 8 mi/h. Her brother, Ted, leaves home on his bicycle 1/2 an hour later, following Beth’s route. He rides at a steady rate of 12 mi/h. How long after Beth leaves home will Ted catch up? Word Problem - Answer Beth leaves home on her bicycle, riding at a steady rate of 8 mi/h. Her brother, Ted, leaves home on his bicycle 1/2 an hour later, following Beth’s route. He rides at a steady rate of 12 mi/h. How long after Beth leaves home will Ted catch up? 8x = 12(x - 1/2) x = 1.5 7-5 Solving Equations with Variables on both sides (Day 2) 5(w + 3) = 4(w - 2) 9 - d = -24 - 4d 7-5 Solving Equations with Variables on both sides (Day 2) - Answers 5(w + 3) = 4(w - 2) w= -23 9 - d = -24 - 4d d= -11 Word Problems • Five more than three times a number is the same as four less than twice a number. Find the number. • Sixty-seven, decreased by four times a number, is the same as eight times a number, increased by seven. Find the number. Word Problems Answers • Five more than three times a number is the same as four less than twice a number. Find the number. 5 + 3y = 2y – 4; y= -9 • Sixty-seven, decreased by four times a number, is the same as eight times a number, increased by seven. Find the number. 67 – 4a = 8a + 7; a = 5 Find the value of x and the perimeter The square and the triangle have equal perimeters. A. Find the value of x B. Find the perimeter (Square: Side is x-3) (Triangle: Sides are x, x, and 8) Find the value of x and the perimeter The square and the triangle have equal perimeters. A. Find the value of x 4(x - 3) = x + x + 8; B. Find the perimeter x = 10 p = 28 (Square: Side is x-3) (Triangle: Sides are x, x, and 8) Find the missing value The Yellow Bus Company charges $160 plus $80 per hour to rent a bus. The Orange Bus Company charges $200 plus $60 per hour. A. For what number of hours would the companies charge the same? B. What would the charge be for that number of hours? Find the missing value The Yellow Bus Company charges $160 plus $80 per hour to rent a bus. The Orange Bus Company charges $200 plus $60 per hour. A. For what number of hours would the companies charge the same? 160 + 80h = 200 + 60h; h = 2 hours B. What would the charge be for that number of hours? $320 7-7 Transforming Formulas You can use the properties of equality to transform a formula to represent one quantity in terms of another. Transforming in one step Solve the area formula A = lw for l Examples: p = s - c (solve for s) h = k/j (solve for k) Transforming in one step - Answers Solve the area formula A = lw for l l = A/w Examples: p = s - c s = p + c h = k/j k=hxj Using more than one step Solve the formula P = 2L + 2W for L Y = 3/5p - 4 solve for p R = n(C - F) solve for C Using more than one step - Answers Solve the formula P = 2L + 2W for L l = (P-2w)/2 Y = 3/5p - 4 solve for p p = 5/3(y + 4) R = n(C - F) solve for C C = (R + nF)/n